title: Foliation

abstract: In mathematics (differential geometry), a foliation is an equivalence relation on an n-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension p, modeled on the decomposition of the real coordinate space Rn into the cosets x + Rp of the standardly embedded subspace Rp. The equivalence classes are called the leaves of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable (of class Cr), or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class Cr it is usually understood that r ≥ 1 (otherwise, C⁰ is a topological foliation). The number p (the dimension of the leaves) is called the dimension of the foliation and q = n − p is called its codimension. In some papers on general relativity by mathematical physicists, the term foliation (or slicing) is used to describe a situation where the relevant Lorentz manifold (a (p+1)-dimensional spacetime) has been decomposed into hypersurfaces of dimension p, specified as the level sets of a real-valued smooth function (scalar field) whose gradient is everywhere non-zero; this smooth function is moreover usually assumed to be a time function, meaning that its gradient is everywhere time-like, so that its level-sets are all space-like hypersurfaces. In deference to standard mathematical terminology, these hypersurface are often called the leaves (or sometimes slices) of the foliation. Note that while this situation does constitute a codimension-1 foliation in the standard mathematical sense, examples of this type are actually globally trivial; while the leaves of a (mathematical) codimension-1 foliation are always locally the level sets of a function, they generally cannot be expressed this way globally, as a leaf may pass through a local-trivializing chart infinitely many times, and the holonomy around a leaf may also obstruct the existence of a globally-consistent defining functions for the leaves. For example, while the 3-sphere has a famous codimension-1 foliation discovered by Reeb, a codimension-1 foliation of a closed manifold cannot be given by the level sets of a smooth function, since a smooth function on a closed manifold necessarily has critical points at its maxima and minima.

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# Foliation

In mathematics (differential geometry), a **foliation** is an equivalence relation on an *n*-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension *p*, modeled on the decomposition of the real coordinate space **R***n* into the cosets *x* \+ **R***p* of the standardly embedded subspace **R***p*. The equivalence classes are called the **leaves** of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable (of class *Cr*), or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class *Cr* it is usually understood that *r* ≥ 1 (otherwise, *C*0 is a topological foliation). The number *p* (the dimension of the leaves) is called the dimension of the foliation and *q* = *n**p* is called its codimension.

In some papers on general relativity by mathematical physicists, the term foliation (or **slicing**) is used to describe a situation where the relevant Lorentz manifold (a (*p*+1)-dimensional spacetime) has been decomposed into hypersurfaces of dimension *p*, specified as the level sets of a real-valued smooth function (scalar field) whose gradient is everywhere non-zero; this smooth function is moreover usually assumed to be a **time function,** meaning that its gradient is everywhere time-like, so that its level-sets are all space-like hypersurfaces. In deference to standard mathematical terminology, these hypersurface are often called the leaves (or sometimes **slices**) of the foliation. Note that while this situation does constitute a codimension-1 foliation in the standard mathematical sense, examples of this type are actually globally trivial; while the leaves of a (mathematical) codimension-1 foliation are always **locally** the level sets of a function, they generally cannot be expressed this way globally, as a leaf may pass through a local-trivializing chart infinitely many times, and the holonomy around a leaf may also obstruct the existence of a globally-consistent defining functions for the leaves. For example, while the 3-sphere has a famous codimension-1 foliation discovered by Reeb, a codimension-1 foliation of a closed manifold cannot be given by the level sets of a smooth function, since a smooth function on a closed manifold necessarily has critical points at its maxima and minima.

## Foliated charts and atlases

In order to give a more precise definition of foliation, it is necessary to define some auxiliary elements.

A *rectangular neighborhood* in **R***n* is an open subset of the form *B* = *J*1 × ⋅⋅⋅ × *J*n, where *Ji* is a (possibly unbounded) relatively open interval in the *i*th coordinate axis. If *J*1 is of the form (*a*,0\], it is said that *B* has boundary

$$\partial B=\left\\{\left(0,x^{2},\ldots ,x^{n}\right)\in B\right\\}.$$

In the following definition, coordinate charts are considered that have values in **R***p* × **R***q*, allowing the possibility of manifolds with boundary and (convex) corners.

A *foliated chart* on the *n*-manifold *M* of codimension *q* is a pair (*U*,*φ*), where *U**M* is open and $\varphi :U\to B\_{\tau }\times B\_{\pitchfork }$ is a diffeomorphism, $B\_{\pitchfork }$ being a rectangular neighborhood in **R***q* and $B\_{\tau }$ a rectangular neighborhood in **R***p*. The set *Py* = *φ*−1(*Bτ* × {*y*}), where $y\in B\_{\pitchfork }$, is called a *plaque* of this foliated chart. For each x ∈ *Bτ*, the set *Sx* = *φ*−1({*x*} × $B\_{\pitchfork }$) is called a *transversal* of the foliated chart. The set *τU* = *φ*−1(*Bτ* × ( $B\_{\pitchfork }$)) is called the *tangential boundary* of *U* and $\partial \_{\pitchfork }U$ = *φ*−1((*∂Bτ*) × $B\_{\pitchfork }$) is called the *transverse boundary* of *U*.

The foliated chart is the basic model for all foliations, the plaques being the leaves. The notation *Bτ* is read as "*B*-tangential" and $B\_{\pitchfork }$ as "*B*-transverse". There are also various possibilities. If both $B\_{\pitchfork }$ and *Bτ* have empty boundary, the foliated chart models codimension-*q* foliations of *n*-manifolds without boundary. If one, but not both of these rectangular neighborhoods has boundary, the foliated chart models the various possibilities for foliations of *n*-manifolds with boundary and without corners. Specifically, if  $B\_{\pitchfork }$ ≠ ∅ = *∂Bτ*, then *∂U* = *τU* is a union of plaques and the foliation by plaques is tangent to the boundary. If *∂Bτ* ≠ ∅ =  $B\_{\pitchfork }$, then *∂U* = $\partial \_{\pitchfork }U$ is a union of transversals and the foliation is transverse to the boundary. Finally, if  $B\_{\pitchfork }$ ≠ ∅ ≠ *∂Bτ*, this is a model of a foliated manifold with a corner separating the tangential boundary from the transverse boundary.

A *foliated atlas* of codimension *q* and class *Cr* (0 ≤ *r* ≤ ∞) on the *n*-manifold *M* is a *Cr*-atlas $\mathcal {U}=\\{(U\_{\alpha },\varphi \_{\alpha })\mid \alpha \in A\\}$ of foliated charts of codimension *q* which are *coherently foliated* in the sense that, whenever *P* and *Q* are plaques in distinct charts of $\mathcal {U}$, then *P**Q* is open both in *P* and *Q*.

A useful way to reformulate the notion of coherently foliated charts is to write for *w**U*α*U*β

$$\varphi \_{\alpha }(w)=\left(x\_{\alpha }(w),y\_{\alpha }(w)\right)\in B\_{\tau }^{\alpha }\times B\_{\pitchfork }^{\alpha },$$

$$\varphi \_{\beta }(w)=\left(x\_{\beta }(w),y\_{\beta }(w)\right)\in B\_{\tau }^{\beta }\times B\_{\pitchfork }^{\beta }.$$

The notation (*U*α,*φ*α) is often written (*U*α,*x*α,*y*α), with

$$x\_{\alpha }=\left(x\_{\alpha }^{1},\dots ,x\_{\alpha }^{p}\right),$$

$$y\_{\alpha }=\left(y\_{\alpha }^{1},\dots ,y\_{\alpha }^{q}\right).$$

On *φ*β(*U*α*U*β) the coordinates formula can be changed as

$$g\_{\alpha \beta }\left(x\_{\beta },y\_{\beta }\right)=\varphi \_{\alpha }\circ \varphi \_{\beta }^{-1}\left(x\_{\beta },y\_{\beta }\right)=\left(x\_{\alpha }\left(x\_{\beta },y\_{\beta }\right),y\_{\alpha }\left(x\_{\beta },y\_{\beta }\right)\right).$$

The condition that (*U**α*,*x**α*,*y**α*) and (*U**β*,*x**β*,*y**β*) be coherently foliated means that, if *P**U**α* is a plaque, the connected components of *P**U**β* lie in (possibly distinct) plaques of *U**β*. Equivalently, since the plaques of *U**α* and *U**β* are level sets of the transverse coordinates *y**α* and *y**β*, respectively, each point *z**U**α**U**β* has a neighborhood in which the formula

$$y\_{\alpha }=y\_{\alpha }(x\_{\beta },y\_{\beta })=y\_{\alpha }(y\_{\beta })$$

is independent of *x**β*.

The main use of foliated atlases is to link their overlapping plaques to form the leaves of a foliation. For this and other purposes, the general definition of foliated atlas above is a bit clumsy. One problem is that a plaque of (*U*α,*φ*α) can meet multiple plaques of (*U*β,*φ*β). It can even happen that a plaque of one chart meets infinitely many plaques of another chart. However, no generality is lost in assuming the situation to be much more regular as shown below.

Two foliated atlases $\mathcal {U}$ and $\mathcal {V}$ on *M* of the same codimension and smoothness class *Cr* are *coherent* $\left({\mathcal {U}\thickapprox {\mathcal {V}\right)}}$ if $\mathcal {U}\cup {\mathcal {V}}$ is a foliated *Cr*-atlas. Coherence of foliated atlases is an equivalence relation.

Plaques and transversals defined above on open sets are also open. But one can speak also of closed plaques and transversals. Namely, if (*U*,*φ*) and (*W*,*ψ*) are foliated charts such that $\overline {U}$ (the closure of *U*) is a subset of *W* and *φ* = *ψ*|*U* then, if $\varphi (U)=B\_{\tau }\times B\_{\pitchfork },$ it can be seen that $\psi |{\overline {U}}$, written $\overline {\varphi }$, carries $\overline {U}$ diffeomorphically onto $\overline {B}\_{\tau }\times {\overline {B}\_{\pitchfork }.}$

A foliated atlas is said to be *regular* if

1. for each α ∈ *A*, $\overline {U}\_{\alpha }$ is a compact subset of a foliated chart (*W*α,*ψ*α) and *φ*α = *ψ*α|*U*α;
2. the cover {*U*α | α ∈ *A*} is locally finite;
3. if (*U*α,*φ*α) and (*U*β,*φ*β) are elements of the foliated atlas, then the interior of each closed plaque *P* ⊂ $\overline {U}\_{\alpha }$ meets at most one plaque in $\overline {U}\_{\beta }.$

By property (1), the coordinates *x*α and *y*α extend to coordinates $\overline {x}\_{\alpha }$ and $\overline {y}\_{\alpha }$ on $\overline {U}\_{\alpha }$ and one writes $\overline {\varphi }\_{\alpha }=\left({\overline {x}\_{\alpha },{\overline {y}\_{\alpha }\right).}}$ Property (3) is equivalent to requiring that, if *U*α*U*β ≠ ∅, the transverse coordinate changes $\overline {y}\_{\alpha }={\overline {y}\_{\alpha }\left({\overline {x}\_{\beta },{\overline {y}\_{\beta }\right)}}}$ be independent of $\overline {x}\_{\beta }.$ That is

$$\overline {g}\_{\alpha \beta }={\overline {\varphi }\_{\alpha }\circ {\overline {\varphi }\_{\beta }^{-1}:{\overline {\varphi }\_{\beta }\left({\overline {U}\_{\alpha }\cap {\overline {U}\_{\beta }\right)\rightarrow {\overline {\varphi }\_{\alpha }\left({\overline {U}\_{\alpha }\cap {\overline {U}\_{\beta }\right)}}}}}}}}$$

has the formula

$$\overline {g}\_{\alpha \beta }\left({\overline {x}\_{\beta },{\overline {y}\_{\beta }\right)=\left({\overline {x}\_{\alpha }\left({\overline {x}\_{\beta },{\overline {y}\_{\beta }\right),{\overline {y}\_{\alpha }\left({\overline {y}\_{\beta }\right)\right).}}}}}}}$$

Similar assertions hold also for open charts (without the overlines). The transverse coordinate map *y*α can be viewed as a submersion

$$y\_{\alpha }:U\_{\alpha }\rightarrow \mathbb {R} ^{q}$$

and the formulas *y*α = *y*α(*y*β) can be viewed as diffeomorphisms

$$\gamma \_{\alpha \beta }:y\_{\beta }\left(U\_{\alpha }\cap U\_{\beta }\right)\rightarrow y\_{\alpha }\left(U\_{\alpha }\cap U\_{\beta }\right).$$

These satisfy the cocycle conditions. That is, on *y*δ(*U*α*U*β*U*δ),

$$\gamma \_{\alpha \delta }=\gamma \_{\alpha \beta }\circ \gamma \_{\beta \delta }$$

and, in particular,

$$\gamma \_{\alpha \alpha }\equiv y\_{\alpha }\left(U\_{\alpha }\right),$$

$$\gamma \_{\alpha \beta }=\gamma \_{\beta \alpha }^{-1}.$$

Using the above definitions for coherence and regularity it can be proven that every foliated atlas has a coherent refinement that is regular.

## Foliation definitions

Several alternative definitions of foliation exist depending on the way through which the foliation is achieved. The most common way to achieve a foliation is through decomposition reaching to the following

**Definition.** A *p*-dimensional, class *Cr* foliation of an *n*-dimensional manifold *M* is a decomposition of *M* into a union of disjoint connected submanifolds {*L*α}α∈*A*, called the *leaves* of the foliation, with the following property: Every point in *M* has a neighborhood *U* and a system of local, class *Cr* coordinates *x*=(*x*1, ⋅⋅⋅, *xn*) : *U***R***n* such that for each leaf *L*α, the components of *U**L*α are described by the equations *x**p*+1=constant, ⋅⋅⋅, *xn*=constant. A foliation is denoted by $\mathcal {F}$={*L*α}α∈*A*.

The notion of leaves allows for an intuitive way of thinking about a foliation. For a slightly more geometrical definition, p-dimensional foliation $\mathcal {F}$ of an n-manifold M may be thought of as simply a collection {*Ma*}  of pairwise-disjoint, connected, immersed p-dimensional submanifolds (the leaves of the foliation) of M, such that for every point x in M, there is a chart $(U,\varphi )$ with U homeomorphic to **R***n* containing x such that every leaf, *Ma*, meets U in either the empty set or a countable collection of subspaces whose images under $\varphi $ in $\varphi (M\_{a}\cap U)$ are p-dimensional affine subspaces whose first *n**p* coordinates are constant.

Locally, every foliation is a submersion allowing the following

**Definition.** Let *M* and *Q* be manifolds of dimension *n* and *q**n* respectively, and let *f* : *M**Q* be a submersion, that is, suppose that the rank of the function differential (the Jacobian) is *q*. It follows from the Implicit Function Theorem that *ƒ* induces a codimension-*q* foliation on *M* where the leaves are defined to be the components of *f*−1(*x*) for *x**Q*.

This definition describes a dimension-p foliation $\mathcal {F}$ of an n-dimensional manifold M that is a covered by charts *Ui* together with maps

$$\varphi \_{i}:U\_{i}\to \mathbb {R} ^{n}$$

such that for overlapping pairs *Ui*, *Uj* the transition functions *φij* : **R***n***R***n* defined by

$$\varphi \_{ij}=\varphi \_{j}\varphi \_{i}^{-1}$$

take the form

$$\varphi \_{ij}(x,y)=(\varphi \_{ij}^{1}(x),\varphi \_{ij}^{2}(x,y))$$

where x denotes the first q = *n**p* coordinates, and y denotes the last p co-ordinates. That is,

$$\begin{aligned}\varphi \_{ij}^{1}:{}&\mathbb {R} ^{q}\to \mathbb {R} ^{q}\\\varphi \_{ij}^{2}:{}&\mathbb {R} ^{n}\to \mathbb {R} ^{p}\end{aligned}$$

The splitting of the transition functions *φij* into $\varphi \_{ij}^{1}(x)$ and $\varphi \_{ij}^{2}(x,y)$ as a part of the submersion is completely analogous to the splitting of $\overline {g}\_{\alpha \beta }$ into $\overline {y}\_{\alpha }\left({\overline {y}\_{\beta }\right)}$ and $\overline {x}\_{\alpha }\left({\overline {x}\_{\beta },{\overline {y}\_{\beta }\right)}}$ as a part of the definition of a regular foliated atlas. This makes possible another definition of foliations in terms of regular foliated atlases. To this end, one has to prove first that every regular foliated atlas of codimension *q* is associated to a unique foliation $\mathcal {F}$ of codimension *q*.

As shown in the proof, the leaves of the foliation are equivalence classes of plaque chains of length ≤ *p* which are also topologically immersed Hausdorff p-dimensional submanifolds. Next, it is shown that the equivalence relation of plaques on a leaf is expressed in equivalence of coherent foliated atlases in respect to their association with a foliation. More specifically, if $\mathcal {U}$ and $\mathcal {V}$ are foliated atlases on *M* and if $\mathcal {U}$ is associated to a foliation $\mathcal {F}$ then $\mathcal {U}$ and $\mathcal {V}$ are coherent if and only if $\mathcal {V}$ is also associated to $\mathcal {F}$.

It is now obvious that the correspondence between foliations on *M* and their associated foliated atlases induces a one-to-one correspondence between the set of foliations on *M* and the set of coherence classes of foliated atlases or, in other words, a foliation $\mathcal {F}$ of codimension *q* and class *Cr* on *M* is a coherence class of foliated atlases of codimension *q* and class *Cr* on *M*. By Zorn's lemma, it is obvious that every coherence class of foliated atlases contains a unique maximal foliated atlas. Thus,

**Definition.** A foliation of codimension *q* and class *Cr* on *M* is a maximal foliated *Cr*-atlas of codimension *q* on *M*.

In practice, a relatively small foliated atlas is generally used to represent a foliation. Usually, it is also required this atlas to be regular.

In the chart *Ui*, the stripes *x* = constant match up with the stripes on other charts *Uj*. These submanifolds piece together from chart to chart to form maximal connected injectively immersed submanifolds called the **leaves** of the foliation.

If one shrinks the chart *Ui* it can be written as *Uix* × *Uiy*, where *Uix***R***n**p*, *Uiy***R***p*, *Uiy* is homeomorphic to the plaques, and the points of *Uix* parametrize the plaques in *Ui*. If one picks *y*0 in *Uiy*, then *Uix* × {*y*0}  is a submanifold of *Ui* that intersects every plaque exactly once. This is called a local *transversal* *section* of the foliation. Note that due to monodromy global transversal sections of the foliation might not exist.

The case *r* = 0 is rather special. Those *C*0 foliations that arise in practice are usually "smooth-leaved". More precisely, they are of class *C**r*,0, in the following sense.

**Definition.** A foliation $\mathcal {F}$ is of class *Cr,k*, *r* \> *k* ≥ 0, if the corresponding coherence class of foliated atlases contains a regular foliated atlas {*U*α,*x*α,*y*α}α∈*A* such that the change of coordinate formula

$$g\_{\alpha \beta }(x\_{\beta },y\_{\beta })=(x\_{\alpha }(x\_{\beta },y\_{\beta }),y\_{\alpha }(y\_{\beta })).$$

is of class *Ck*, but *x*α is of class *Cr* in the coordinates *x*β and its mixed *x*β partials of orders ≤ *r* are *Ck* in the coordinates (*x*β,*y*β).

The above definition suggests the more general concept of a *foliated space* or *abstract lamination*. One relaxes the condition that the transversals be open, relatively compact subsets of **R***q*, allowing the transverse coordinates *y*α to take their values in some more general topological space *Z*. The plaques are still open, relatively compact subsets of **R***p*, the change of transverse coordinate formula *y*α(*y*β) is continuous and *x**α*(*x**β*,*y**β*) is of class *Cr* in the coordinates *x**β* and its mixed *x**β* partials of orders ≤ *r* are continuous in the coordinates (*x**β*,*y**β*). One usually requires *M* and *Z* to be locally compact, second countable and metrizable. This may seem like a rather wild generalization, but there are contexts in which it is useful.

## Holonomy

Let (*M*, $\mathcal {F}$) be a foliated manifold. If *L* is a leaf of $\mathcal {F}$ and *s* is a path in *L*, one is interested in the behavior of the foliation in a neighborhood of *s* in *M*. Intuitively, an inhabitant of the leaf walks along the path *s*, keeping an eye on all of the nearby leaves. As they (hereafter denoted by *s*(*t*)) proceed, some of these leaves may "peel away", getting out of visual range, others may suddenly come into range and approach *L* asymptotically, others may follow along in a more or less parallel fashion or wind around *L* laterally, *etc*. If *s* is a loop, then *s*(*t*) repeatedly returns to the same point *s*(*t*0) as *t* goes to infinity and each time more and more leaves may have spiraled into view or out of view, *etc*. This behavior, when appropriately formalized, is called the *holonomy* of the foliation.

Holonomy is implemented on foliated manifolds in various specific ways: the total holonomy group of foliated bundles, the holonomy pseudogroup of general foliated manifolds, the germinal holonomy groupoid of general foliated manifolds, the germinal holonomy group of a leaf, and the infinitesimal holonomy group of a leaf.

### Foliated bundles

The easiest case of holonomy to understand is the *total holonomy* of a foliated bundle. This is a generalization of the notion of a *Poincaré map*.

The term "first return (recurrence) map" comes from the theory of dynamical systems. Let Φ*t* be a nonsingular *Cr* flow (*r*  1) on the compact *n*-manifold *M*. In applications, one can imagine that *M* is a cyclotron or some closed loop with fluid flow. If *M* has a boundary, the flow is assumed to be tangent to the boundary. The flow generates a 1-dimensional foliation $\mathcal {F}$. If one remembers the positive direction of flow, but otherwise forgets the parametrization (shape of trajectory, velocity, *etc*.), the underlying foliation $\mathcal {F}$ is said to be oriented. Suppose that the flow admits a global cross section *N*. That is, *N* is a compact, properly embedded, *Cr* submanifold of *M* of dimension *n*  1, the foliation $\mathcal {F}$ is transverse to *N*, and every flow line meets *N*. Because the dimensions of *N* and of the leaves are complementary, the transversality condition is that

$$T\_{y}(M)=T\_{y}({\mathcal {F})\oplus T\_{y}(N){\text{ for each }y\in N.}}$$

Let *y**N* and consider the *ω*-limit set ω(*y*) of all accumulation points in *M* of all sequences $\left\\{\Phi \_{k}(y)\right\\}\_{k=1}^{\infty }$, where *tk* goes to infinity. It can be shown that ω(y) is compact, nonempty, and a union of flow lines. If $z=\lim \_{k\rightarrow \infty }\Phi \_{k}\in \omega (y),$ there is a value *t** ∈ **R** such that Φ*t**(*z*) ∈ *N* and it follows that

$$\lim \_{k\to \infty }\Phi \_{k}+t^{\ast }(y)=\Phi \_{t^{\ast }(z)\in N.}$$

Since *N* is compact and $\mathcal {F}$ is transverse to *N*, it follows that the set {*t* \> 0 | Φ*t*(*y*) ∈ N} is a monotonically increasing sequence $\\{\tau \_{k}(y)\\}\_{k=1}^{\infty }$ that diverges to infinity.

As *y**N* varies, let *τ*(*y*) = *τ*1(*y*), defining in this way a positive function *τ**Cr*(*N*) (the first return time) such that, for arbitrary *y**N*, Φ*t*(*y*) ∉ *N*, 0 \< *t* \< *τ*(*y*), and Φ*τ*(*y*)(*y*) ∈ *N*.

Define *f* : *N**N* by the formula *f*(*y*) = Φ*τ*(*y*)(*y*). This is a *C**r* map. If the flow is reversed, exactly the same construction provides the inverse *f*−1; so *f* ∈ Diff*r*(*N*). This diffeomorphism is the first return map and τ is called the *first return time*. While the first return time depends on the parametrization of the flow, it should be evident that *f* depends only on the oriented foliation $\mathcal {F}$. It is possible to reparametrize the flow Φ*t*, keeping it nonsingular, of class *Cr*, and not reversing its direction, so that *τ*  1.

The assumption that there is a cross section N to the flow is very restrictive, implying that *M* is the total space of a fiber bundle over *S*1. Indeed, on **R** × *N*, define ~*f* to be the equivalence relation generated by

$$(t,y)\sim \_{f}(t-1,f(y)).$$

Equivalently, this is the orbit equivalence for the action of the additive group **Z** on **R** × *N* defined by

$$k\cdot (t,y)=(t-k,f^{k}(y)),$$

for each *k***Z** and for each (*t*,*y*) ∈ **R** × *N*. The mapping cylinder of *f* is defined to be the *C**r* manifold

$$M\_{f}=(\mathbb {R} \times N)/{\sim \_{f}.}$$

By the definition of the first return map *f* and the assumption that the first return time is τ ≡ 1, it is immediate that the map

$$\Phi :\mathbb {R} \times N\rightarrow M.$$

defined by the flow, induces a canonical *C**r* diffeomorphism

$$\varphi :M\_{f}\rightarrow M.$$

If we make the identification *M**f* = *M*, then the projection of **R** × *N* onto **R** induces a *C**r* map

$$\pi :M\rightarrow \mathbb {R} /\mathbb {Z} =S^{1}$$

that makes *M* into the total space of a fiber bundle over the circle. This is just the projection of *S*1 × *D*2 onto *S*1. The foliation $\mathcal {F}$ is transverse to the fibers of this bundle and the bundle projection π, restricted to each leaf *L*, is a covering map π : *L**S*1. This is called a *foliated bundle*.

Take as basepoint *x*0*S*1 the equivalence class 0 + **Z**; so π−1(*x*0) is the original cross section *N*. For each loop *s* on *S*1, based at *x*0, the homotopy class \[*s*\] ∈ π1(*S*1,*x*0) is uniquely characterized by deg *s***Z**. The loop *s* lifts to a path in each flow line and it should be clear that the lift *sy* that starts at *y**N* ends at *fk*(*y*) ∈ *N*, where *k* = deg *s*. The diffeomorphism *fk* ∈ Diff*r*(*N*) is also denoted by *hs* and is called the *total holonomy* of the loop *s*. Since this depends only on \[*s*\], this is a definition of a homomorphism

$$h:\pi \_{1}(S^{1},x\_{0})\rightarrow \operatorname {Diff} ^{\\,r}(N),$$

called the *total holonomy homomorphism* for the foliated bundle.

Using fiber bundles in a more direct manner, let (*M*, $\mathcal {F}$) be a foliated *n*-manifold of codimension *q*. Let π : *M**B* be a fiber bundle with *q*-dimensional fiber *F* and connected base space *B*. Assume that all of these structures are of class *Cr*, 0 ≤ *r* ≤ ∞, with the condition that, if *r* = 0, *B* supports a *C*1 structure. Since every maximal *C*1 atlas on *B* contains a *C* subatlas, no generality is lost in assuming that *B* is as smooth as desired. Finally, for each *x**B*, assume that there is a connected, open neighborhood *U**B* of *x* and a local trivialization

$$\begin{matrix}\pi ^{-1}(U)&{\xrightarrow {\varphi }&U\times {F}\\\scriptstyle {\pi }{\Bigg \downarrow }&{\qquad }&{\Bigg \downarrow }{\scriptstyle {p}\\U&{\xrightarrow {\text{id}&U\end{matrix}}}}}$$

where *φ* is a *Cr* diffeomorphism (a homeomorphism, if *r* = 0) that carries $\textstyle {\mathcal {F}\mid \pi ^{-1}(U)}$ to the product foliation {*U* × {*y*}}*y*  *F*. Here, $\textstyle {\mathcal {F}\mid \pi ^{-1}(U)}$ is the foliation with leaves the connected components of *L* ∩ π−1(*U*), where *L* ranges over the leaves of $\mathcal {F}$. This is the general definition of the term "foliated bundle" (*M*, $\mathcal {F}$,π) of class *Cr*.

$\mathcal {F}$ is transverse to the fibers of π (it is said that $\mathcal {F}$ is transverse to the fibration) and that the restriction of π to each leaf *L* of $\mathcal {F}$ is a covering map π : *L**B*. In particular, each fiber *Fx* = π−1(*x*) meets every leaf of $\mathcal {F}$. The fiber is a cross section of $\mathcal {F}$ in complete analogy with the notion of a cross section of a flow.

The foliation $\mathcal {F}$ being transverse to the fibers does not, of itself, guarantee that the leaves are covering spaces of *B*. A simple version of the problem is a foliation of **R**2, transverse to the fibration

$$\pi :\mathbb {R} ^{2}\rightarrow \mathbb {R} ,$$

$$\pi (x,y)=x,$$

but with infinitely many leaves missing the *y*-axis. In the respective figure, it is intended that the "arrowed" leaves, and all above them, are asymptotic to the axis *x* = 0. One calls such a foliation incomplete relative to the fibration, meaning that some of the leaves "run off to infinity" as the parameter *x**B* approaches some *x*0*B*. More precisely, there may be a leaf *L* and a continuous path *s* : \[0,*a*) → *L* such that lim*t**a*π(*s*(*t*)) = *x*0*B*, but lim*t**a**s*(*t*) does not exist in the manifold topology of *L*. This is analogous to the case of incomplete flows, where some flow lines "go to infinity" in finite time. Although such a leaf *L* may elsewhere meet π−1(*x*0), it cannot evenly cover a neighborhood of *x*0, hence cannot be a covering space of *B* under π. When *F* is compact, however, it is true that transversality of $\mathcal {F}$ to the fibration does guarantee completeness, hence that $\textstyle (M,{\mathcal {F},\pi )}$ is a foliated bundle.

There is an atlas $\mathcal {U}$ = {*U*α,*x*α}α∈A on *B*, consisting of open, connected coordinate charts, together with trivializations *φ**α* : *π*−1(*U**α*) → *U**α* × *F* that carry $\mathcal {F}$|π−1(*U**α*) to the product foliation. Set *W**α* = *π*−1(*U**α*) and write *φ**α* = (*x**α*,*y**α*) where (by abuse of notation) *x*α represents *x**α**π* and *y**α* : *π*−1(*U*α) → *F* is the submersion obtained by composing *φ*α with the canonical projection *U*α × *F**F*.

The atlas $\mathcal {W}$ = {*W**α*,*x**α*,*y**α*}*α**A* plays a role analogous to that of a foliated atlas. The plaques of *W**α* are the level sets of *y**α* and this family of plaques is identical to *F* via *y**α*. Since *B* is assumed to support a *C* structure, according to the Whitehead theorem one can fix a Riemannian metric on *B* and choose the atlas $\mathcal {U}$ to be geodesically convex. Thus, *U*α*U**β* is always connected. If this intersection is nonempty, each plaque of *W**α* meets exactly one plaque of *W**β*. Then define a *holonomy cocycle* $\gamma =\left\\{\gamma \_{\alpha \beta }\right\\}\_{\alpha ,\beta \in A}$ by setting

$$\gamma \_{\alpha \beta }=y\_{\alpha }\circ y\_{\beta }^{-1}:F\rightarrow F.$$

## Examples

### Flat space

Consider an n-dimensional space, foliated as a product by subspaces consisting of points whose first *n**p* coordinates are constant. This can be covered with a single chart. The statement is essentially that **R***n* = **R***n**p* × **R***p* with the leaves or plaques **R***p* being enumerated by **R***n**p*. The analogy is seen directly in three dimensions, by taking *n* = 3 and *p* = 2: the 2-dimensional leaves of a book are enumerated by a (1-dimensional) page number.

### Bundles

A rather trivial example of foliations are products *M* = *B* × *F*, foliated by the leaves *Fb* = {*b*} × *F*, *b**B*. (Another foliation of M is given by *Bf* = *B* × {*f* } , *f* *F*.)

A more general class are flat G-bundles with *G* = Homeo(*F*) for a manifold F. Given a representation *ρ* : π1(*B*) → Homeo(*F*), the flat Homeo(*F*)-bundle with monodromy ρ is given by $M=\left({\widetilde {B}\times F\right)/\pi \_{1}B}$, where π1(*B*) acts on the universal cover $\widetilde {B}$ by deck transformations and on F by means of the representation ρ.

Flat bundles fit into the framework of fiber bundles. A map π : *M**B* between manifolds is a fiber bundle if there is a manifold F such that each *b**B* has an open neighborhood U such that there is a homeomorphism $\varphi :\pi ^{-1}(U)\to U\times F$ with $\pi =p\_{1}\varphi $, with *p*1 : *U* × *F**U* projection to the first factor. The fiber bundle yields a foliation by fibers $F\_{b}:=\pi ^{-1}(\\{b\\}),b\in B$. Its space of leaves L is homeomorphic to B, in particular L is a Hausdorff manifold.

### Coverings

If *M**N* is a covering map between manifolds, and F is a foliation on N, then it pulls back to a foliation on M. More generally, if the map is merely a branched covering, where the branch locus is transverse to the foliation, then the foliation can be pulled back.

### Submersions

If *Mn**Nq,* (*q**n*) is a submersion of manifolds, it follows from the inverse function theorem that the connected components of the fibers of the submersion define a codimension q foliation of M. Fiber bundles are an example of this type.

An example of a submersion, which is not a fiber bundle, is given by

$$\begin{cases}f:\[-1,1\]\times \mathbb {R} \to \mathbb {R} \\f(x,y)=(x^{2}-1)e^{y}\end{cases}$$

This submersion yields a foliation of \[−1, 1\] × **R** which is invariant under the **Z**-actions given by

$$z(x,y)=(x,y+n),\quad {\text{or}\quad z(x,y)=\left((-1)^{n}x,y\right)}$$

for (*x*, *y*) ∈ \[−1, 1\] × **R** and *n***Z**. The induced foliations of **Z** \ (\[−1, 1\] × **R**) are called the 2-dimensional Reeb foliation (of the annulus) resp. the 2-dimensional nonorientable Reeb foliation (of the Möbius band). Their leaf spaces are not Hausdorff.

### Reeb foliations

Define a submersion

$$\begin{cases}f:D^{n}\times \mathbb {R} \to \mathbb {R} \\f(r,\theta ,t):=(r^{2}-1)e^{t}\end{cases}$$

where (*r*, *θ*) ∈ \[0, 1\] × *S**n*−1 are cylindrical coordinates on the n-dimensional disk *Dn*. This submersion yields a foliation of *Dn* × **R** which is invariant under the **Z**-actions given by

$$z(x,y)=(x,y+z)$$

for (*x*, *y*) ∈ *Dn* × **R**, *z***Z**. The induced foliation of **Z** \ (*Dn* × **R**) is called the n-dimensional Reeb foliation. Its leaf space is not Hausdorff.

For *n* = 2, this gives a foliation of the solid torus which can be used to define the Reeb foliation of the 3-sphere by gluing two solid tori along their boundary. Foliations of odd-dimensional spheres *S*2*n*+1 are also explicitly known.

### Lie groups

If G is a Lie group, and H is a Lie subgroup, then G is foliated by cosets of H. When H is closed in G, the quotient space G/H is a smooth (Hausdorff) manifold turning G into a fiber bundle with fiber H and base G/H. This fiber bundle is actually principal, with structure group H.

### Lie group actions

Let G be a Lie group acting smoothly on a manifold M. If the action is a locally free action or free action, then the orbits of G define a foliation of M.

### Linear and Kronecker foliations

If $\tilde {X}$ is a nonsingular (*i.e.*, nowhere zero) vector field, then the local flow defined by $\tilde {X}$ patches together to define a foliation of dimension 1. Indeed, given an arbitrary point *x**M*, the fact that $\tilde {X}$ is nonsingular allows one to find a coordinate neighborhood (*U*,*x*1,...,*xn*) about *x* such that

$$-\varepsilon \<x^{i}\<\varepsilon ,\quad 1\leq i\leq n,$$

and

$$\frac {\partial }{\partial x^{1}={\tilde {X}\mid U.}}$$

Geometrically, the flow lines of $\tilde {X}\mid U$ are just the level sets

$$x^{i}=c^{i},\quad 2\leq i\leq n,$$

where all $|c^{i}|\<\varepsilon .$ Since by convention manifolds are second countable, leaf anomalies like the "long line" are precluded by the second countability of *M* itself. The difficulty can be sidestepped by requiring that $\tilde {X}$ be a complete field (*e.g.*, that *M* be compact), hence that each leaf be a flow line.

An important class of 1-dimensional foliations on the torus *T*2 are derived from projecting constant vector fields on *T*2. A constant vector field

$$\tilde {X}\equiv {\begin{bmatrix}a\\b\end{bmatrix}}$$

on **R**2 is invariant by all translations in **R**2, hence passes to a well-defined vector field *X* when projected on the torus *T*2= **R**2/**Z**2. It is assumed that *a* ≠ 0. The foliation $\mathcal {\tilde {F}}$ on **R**2 produced by $\tilde {X}$ has as leaves the parallel straight lines of slope θ = *b*/*a*. This foliation is also invariant under translations and passes to the foliation $\mathcal {F}$ on *T*2 produced by *X*.

Each leaf of $\mathcal {\tilde {F}}$ is of the form

$$\tilde {L}=\\{(x\_{0}+ta,y\_{0}+tb)\\}\_{t\in \mathbb {R} }.$$

If the slope is rational then all leaves are closed curves homeomorphic to the circle. In this case, one can take *a*,*b***Z**. For fixed *t***R**, the points of $\tilde {L}$ corresponding to values of *t**t*0 \+ **Z** all project to the same point of *T*2; so the corresponding leaf *L* of $\mathcal {F}$ is an embedded circle in *T*2. Since *L* is arbitrary, $\mathcal {F}$ is a foliation of *T*2 by circles. It follows rather easily that this foliation is actually a fiber bundle π : *T*2*S*1. This is known as a *linear foliation*.

When the slope θ = *b*/*a* is irrational, the leaves are noncompact, homeomorphic to the non-compactified real line, and dense in the torus (cf Irrational rotation). The trajectory of each point (*x*0,*y*0) never returns to the same point, but generates an "everywhere dense" winding about the torus, i.e. approaches arbitrarily close to any given point. Thus the closure to the trajectory is the entire two-dimensional torus. This case is named *Kronecker foliation*, after Leopold Kronecker and his

**Kronecker's Density Theorem**. If the real number θ is distinct from each rational multiple of π, then the set {*einθ* | *n***Z**} is dense in the unit circle.

A similar construction using a foliation of **R***n* by parallel lines yields a 1-dimensional foliation of the n-torus **R***n*/**Z***n* associated with the linear flow on the torus.

### Suspension foliations

A flat bundle has not only its foliation by fibres but also a foliation transverse to the fibers, whose leaves are

$$L\_{f}:=\left\\{p\left({\tilde {b},f\right):{\tilde {b}\in {\widetilde {B}\right\\},\quad {\mbox{ for }f\in F,}}}}$$

where $p:{\widetilde {B}\times F\to M}$ is the canonical projection. This foliation is called the suspension of the representation *ρ* : π1(*B*) → Homeo(*F*).

In particular, if *B* = *S*1 and $\varphi :F\to F$ is a homeomorphism of F, then the suspension foliation of $\varphi $ is defined to be the suspension foliation of the representation *ρ* : **Z** → Homeo(*F*) given by *ρ*(*z*) = Φ*z*. Its space of leaves is *L* = $\mathcal {F}$/~, where *x* ~ *y* whenever *y* = Φ*n*(*x*) for some *n***Z**.

The simplest example of foliation by suspension is a manifold *X* of dimension *q*. Let *f* : *X**X* be a bijection. One defines the suspension *M* = *S*1 ×*f* *X* as the quotient of \[0,1\] × *X* by the equivalence relation (1,*x*) ~ (0,*f*(*x*)).

*M* = *S*1 ×*f* *X* = \[0,1\] × *X*

Then automatically *M* carries two foliations: $\mathcal {F}$2 consisting of sets of the form *F*2,*t* = {(*t*,*x*)~ : *x**X*} and $\mathcal {F}$1 consisting of sets of the form *F*2,*x*0 = {(*t*,*x*) : *t* ∈ \[0,1\] ,*x* ∈ Ox0}, where the orbit Ox0 is defined as

O*x*0 = {..., *f*−2(*x*0), *f*−1(*x*0), *x*0, *f*(*x*0), *f*2(*x*0), ...},

where the exponent refers to the number of times the function *f* is composed with itself. Note that O*x*0 = O*f*(*x*0) = O*f*−2(*x*0), etc., so the same is true for *F*1,*x*0. Understanding the foliation $\mathcal {F}$1 is equivalent to understanding the dynamics of the map *f*. If the manifold *X* is already foliated, one can use the construction to increase the codimension of the foliation, as long as *f* maps leaves to leaves.

The Kronecker foliations of the 2-torus are the suspension foliations of the rotations *Rα* : *S*1*S*1 by angle *α* ∈ \[0, 2π).

More specifically, if Σ = Σ2 is the two-holed torus with C1,C2 ∈ Σ the two embedded circles let $\mathcal {F}$ be the product foliation of the 3-manifold *M* = Σ × *S*1 with leaves Σ × {*y*}, *y**S*1. Note that *Ni* = *Ci* × *S*1 is an embedded torus and that $\mathcal {F}$ is transverse to *Ni*, *i* = 1,2. Let Diff+(*S*1) denote the group of orientation-preserving diffeomorphisms of *S*1 and choose *f*1,*f*2 ∈ Diff+(*S*1). Cut *M* apart along *N*1 and *N*2, letting $N\_{i}^{+}$ and $N\_{i}^{-}$ denote the resulting copies of *Ni*, *i* = 1,2. At this point one has a manifold *M'* = Σ' × *S*1 with four boundary components $\left\\{N\_{i}^{\pm }\right\\}\_{i=1,2}.$ The foliation $\mathcal {F}$ has passed to a foliation $\mathcal {F^{\prime }}$ transverse to the boundary ∂*M'* , each leaf of which is of the form Σ' × {*y*}, *y**S*1.

This leaf meets ∂*M'*  in four circles $C\_{i}^{\pm }\times \\{y\\}\subset N\_{i}^{\pm }.$ If *z**Ci*, the corresponding points in $C\_{i}^{\pm }$ are denoted by *z*± and $N\_{i}^{-}$ is "reglued" to $N\_{i}^{+}$ by the identification

$$(z^{-},y)\equiv (z^{+},f\_{i}(y)),\quad i=1,2.$$

Since *f*1 and *f*2 are orientation-preserving diffeomorphisms of *S*1, they are isotopic to the identity and the manifold obtained by this regluing operation is homeomorphic to *M*. The leaves of $\mathcal {F^{\prime }}$, however, reassemble to produce a new foliation $\mathcal {F}$(*f*1,*f*2) of *M*. If a leaf *L* of $\mathcal {F}$(*f*1,*f*2) contains a piece Σ' × {*y*0}, then

$$L=\bigcup \_{g\in G}\Sigma ^{\prime }\times \\{g(y\_{0})\\},$$

where *G* ⊂ Diff+(*S*1) is the subgroup generated by {*f*1,*f*2}. These copies of Σ' are attached to one another by identifications

(*z*,*g*(*y*0)) ≡ (*z*+,*f*1(*g*(*y*0))) for each *z**C*1, (*z*,*g*(*y*0)) ≡ (*z*+,*f*2(*g*(*y*0))) for each *z**C*2,

where *g* ranges over *G*. The leaf is completely determined by the *G*-orbit of *y*0*S*1 and can he simple or immensely complicated. For instance, a leaf will be compact precisely if the corresponding *G*-orbit is finite. As an extreme example, if *G* is trivial (*f*1 = *f*2 = idS1), then $\mathcal {F}$(*f*1,*f*2) = $\mathcal {F}$. If an orbit is dense in *S*1, the corresponding leaf is dense in *M*. As an example, if *f*1 and *f*2 are rotations through rationally independent multiples of 2π, every leaf will be dense. In other examples, some leaf *L* has closure $\bar {L}$ that meets each factor {*w*} × *S*1 in a Cantor set. Similar constructions can be made on Σ × *I*, where *I* is a compact, nondegenerate interval. Here one takes *f*1,*f*2 ∈ Diff+(*I*) and, since ∂*I* is fixed pointwise by all orientation-preserving diffeomorphisms, one gets a foliation having the two components of ∂*M* as leaves. When one forms *M'*  in this case, one gets a foliated manifold with corners. In either case, this construction is called the *suspension* of a pair of diffeomorphisms and is a fertile source of interesting examples of codimension-one foliations.

## Foliations and integrability

There is a close relationship, assuming everything is smooth, with vector fields: given a vector field X on M that is never zero, its integral curves will give a 1-dimensional foliation. (i.e. a codimension *n* − 1 foliation).

This observation generalises to the Frobenius theorem, saying that the necessary and sufficient conditions for a distribution (i.e. an *n**p* dimensional subbundle of the tangent bundle of a manifold) to be tangent to the leaves of a foliation, is that the set of vector fields tangent to the distribution are closed under Lie bracket. One can also phrase this differently, as a question of reduction of the structure group of the tangent bundle from GL(*n*) to a reducible subgroup.

The conditions in the Frobenius theorem appear as integrability conditions; and the assertion is that if those are fulfilled the reduction can take place because local transition functions with the required block structure exist. For example, in the codimension 1 case, we can define the tangent bundle of the foliation as ker(*α*), for some (non-canonical) *α* ∈ Ω1 (i.e. a non-zero co-vector field). A given α is integrable iff *α* = 0 everywhere.

There is a global foliation theory, because topological constraints exist. For example, in the surface case, an everywhere non-zero vector field can exist on an orientable compact surface only for the torus. This is a consequence of the Poincaré–Hopf index theorem, which shows the Euler characteristic will have to be 0. There are many deep connections with contact topology, which is the "opposite" concept, requiring that the integrability condition is *never* satisfied.

## Existence of foliations

Haefliger (1970) gave a necessary and sufficient condition for a distribution on a connected non-compact manifold to be homotopic to an integrable distribution. Thurston (1974, 1976) showed that any compact manifold with a distribution has a foliation of the same dimension.

* G-structure – Structure group sub-bundle on a tangent frame bundle
* Haefliger structure – Generalization of a foliation closed under taking pullbacks.
* Lamination – Partitioned topological space
* Reeb foliation – particular foliation of the 3-sphere introduced by the French mathematician Georges Reeb.
* Taut foliation


title: Garrett-Bullock House

abstract: The Garrett-Bullock House is a historic Queen Anne-style house in Columbus, Georgia. It was listed on the National Register of Historic Places in 1980. It was owned by Joseph Garrett, owner of a Broad Street wholesale liquor-and tobacco business known as "Garrett & Sons", and became a boarding house. The Garrett-Bullock House may have been used as a special events center for weddings and parties. It is separately listed in the National Register and also included in the High Uptown Historic District.

text:
# Garrett-Bullock House

The **Garrett-Bullock House** is a historic Queen Anne-style house in Columbus, Georgia. It was listed on the National Register of Historic Places in 1980.

It was owned by Joseph Garrett, owner of a Broad Street wholesale liquor-and tobacco business known as "Garrett & Sons", and became a boarding house.

The Garrett-Bullock House may have been used as a special events center for weddings and parties.

It is separately listed in the National Register and also included in the High Uptown Historic District.

## InfoBox

|  | |
| --- | --- |
| Garrett-Bullock House | |
| U.S. National Register of Historic Places | |
| U.S. Historic district<br>Contributing property | |
|  | |
|  | |
| Location | 1402 2nd Ave., Columbus, Georgia |
| Coordinates | 32°28′21″N 84°59′25″W / 32.47250°N 84.99028°W / 32.47250; -84.99028 |
| Area | less than one acre |
| Built | 1886 |
| Architectural style | Queen Anne |
| Part of | High Uptown Historic District (ID04000669) |
| MPS | Columbus MRA |
| NRHP reference No. | 80001172 |
| Significant dates | |
| Added to NRHP | September 29, 1980 |
| Designated CP | July 7, 2004 |


title: Lubei Subdistrict, Hengshui

abstract: Lubei Subdistrict is a subdistrict of Taocheng District, Hengshui, Hebei, People's Republic of China, covering part of the northern portion of Hengshui's urban core As of 2011, it has 10 residential communities (社区) under its administration.

text:
# Lubei Subdistrict, Hengshui

**Lubei Subdistrict** (Chinese: 路北街道; pinyin: *Lùběi Jiēdào*; lit. 'north of the road') is a subdistrict of Taocheng District, Hengshui, Hebei, People's Republic of China, covering part of the northern portion of Hengshui's urban core As of 2011, it has 10 residential communities (社区) under its administration.

## InfoBox

| Lubei Subdistrict 路北街道 | |
| --- | --- |
| Subdistrict | |
| Coordinates: 37°45′02″N 115°40′40″E / 37.75056°N 115.67778°E / 37.75056; 115.67778 | |
| Country | People's Republic of China |
| Province | Hebei |
| Prefecture-level city | Hengshui |
| District | Taocheng |
| Village-level divisions | 10 residential communities |
| Elevation | 22 m (72 ft) |
| Time zone | UTC+8 (China Standard) |
| Postal code | 053000 |
| Area code | 0318 |


title: The Southern Cross (Argentina)

abstract: The Southern Cross is an Argentine newspaper founded on January 16, 1875, by Patrick Joseph Dillon (1842–1889), a Roman Catholic priest, editor and politician, born in Tuam, east County Galway, Ireland. Fr. Dillon was appointed a deputy for Buenos Aires Province and president of the Presidential Affairs Commission amongst other positions. He was one of the legislators who proposed Buenos Aires as the federal capital. The Southern Cross was the first entirely Roman Catholic English language publication in Buenos Aires, and continues in print to this day on a monthly basis. A downloadable version can be obtained at The Southern Cross website. Published by Editorial Irlandesa S.A., the paper provides readers with a beginners' guide to the Irish language, helping Irish Argentines keep in touch with their cultural heritage.

text:
# The Southern Cross (Argentina)

***The Southern Cross*** is an Argentine newspaper founded on January 16, 1875, by Patrick Joseph Dillon (1842–1889), a Roman Catholic priest, editor and politician, born in Tuam, east County Galway, Ireland. Fr. Dillon was appointed a deputy for Buenos Aires Province and president of the Presidential Affairs Commission amongst other positions. He was one of the legislators who proposed Buenos Aires as the federal capital.

*The Southern Cross* was the first entirely Roman Catholic English language publication in Buenos Aires, and continues in print to this day on a monthly basis. A downloadable version can be obtained at *The Southern Cross* website.

Published by Editorial Irlandesa S.A., the paper provides readers with a beginners' guide to the Irish language, helping Irish Argentines keep in touch with their cultural heritage.

## InfoBox

|  | |
| --- | --- |
| Type | Monthly newspaper |
| Publisher | Dr. Guillermo MacLoughlin Bréard |
| Founded | 1875 |
| Headquarters | Ríobamba 451 (4°C)<br>Buenos Aires |
| Website | https://thesoutherncross.info/ |


title: Holly Henry

abstract: Holly Mae Heinrich, known professionally as Holly Henry, is an American singer-songwriter from Minnesota. She took her stage name when competing in 2013 on the fifth season of The Voice. She subsequently built an international fan base through her YouTube channel and original music releases.

text:
# Holly Henry

**Holly Mae Heinrich** (born March 31, 1994), known professionally as **Holly Henry**, is an American singer-songwriter from Minnesota. She took her stage name when competing in 2013 on the fifth season of *The Voice* (United States edition). She subsequently built an international fan base through her YouTube channel and original music releases.

## Early life and education

Henry is the daughter of Marc Heinrich, a musician, and Becky Heinrich, an art teacher and graphic artist. She attended public schools in the Minneapolis area and graduated from Hopkins High School. While in high school, Henry sang in a sandwich shop gig and posted covers of songs on her YouTube channel, which she began in 2011. Her most popular video is an *a cappella* cover of Eurythmics' "Sweet Dreams (Are Made of This)" with three-part vocal harmonies.

## Career

While taking a gap year before college, Henry's parents encouraged her to audition for *The Voice* at a Chicago, Illinois audition in January 2013. Henry advanced through the open call and subsequent executive auditions and participated in a boot camp during the summer of 2013. Her Blind Audition performance of *The Scientist* got all four chairs to turn. The studio performance release by the show charted on *Billboard* and the live performance video was in the all-time Top 40 for views of *The Voice* performances on YouTube, until its removal by NBC. In the Battle Rounds, she competed against Cilla Chan, singing *Torn*. Henry and Chan were montaged (their vocal performances were not shown in full) and Henry was chosen to move on. In the Knockout Round, Henry competed against Nic Hawk, singing *Creep* by Radiohead. She was sent home, and NBC's *The Voice* (US) subsequently lost 15% (1.5 million) viewers after that episode. Two years after her appearance on *The Voice*, Henry's popularity on YouTube had increased 10-fold.

Henry released her first EP called *The Immigrant* in 2013. The 5 track EP peaked on iTunes at #6 on both the US Pop and Rock Charts, the same position it attained in pre-release. On April 7, 2014, Henry took second place (to Shawn Mendes) in the RyanSeacrest.com top cover competition for her rendition of the hit song *Say Something*. Third place finisher Jamison Murphy later featured Henry on his single "Remember When". On May 6, 2014, the Grow Music Project released *Session Diaries #4*, a video on the Christopher Tyng-led production of Henry's single "Hide and Seek", which was released simultaneously on iTunes. "Hide and Seek" is the soundtrack for the closing credits of the short film *Escapement*, written and produced by Erik A. Candiani. Henry collaborated with Candiani on the KARE11 (Minneapolis TV station) promo for the Sochi Winter Olympics, which was subsequently nominated for two Upper Midwestern Emmy awards, with a win for best Musical Arrangement/Composition. Henry's original song *In the Air* was commissioned for the 2016 film *Intersection: the Story of Josh Grant*.

Henry's live performances began immediately after her Blind Audition on *The Voice*. They have included a Minnesota Music Coalition benefit at the Varsity Theater on February 20, 2014, and the Winter's Over Music Festival on April 25, 2015. YouTube produced a video of her single *Katie* at their YouTube Space in New York, which Henry released on January 19, 2015. Her Indiegogo-funded second EP, *The Orchard*, was released on August 21, 2015. Henry collaborated with Moonhead Industries to release a music video for *Better*, a single from the EP.

An 11-track album titled *King Paten* was released on July 1, 2016. It had worldwide sales, charting on iTunes in Russia, South Africa, the UK, Cyprus, Israel, France, Canada and the US. On September 2, 2016, she confirmed to writing a new album. On November 17, 2018, Holly released a song called "Float" under the name "Mooney". Again billed as Mooney, she released her self-produced album "softer" on October 25, 2019.

## Reviews

Jonathan Frahm reviewed *The Immigrant* EP and wrote that Henry "displays the beyond-her-years nuance and musicality...delivering what is surely one of the most magnificent releases by a Voice contestant yet." Frahm followed up with a review of "Hide and Seek" in which he wrote about Henry's songwriting: "She’s writing for all to hear, sure, but each song that Holly writes is its own little world in which she is the only initial representative of its populous. It shows, too, in “Hide and Seek”, as she thoughtfully, mournfully weaves a new tale with regards to a wispy nostalgic quality which she so greatly inhabits. “Hide and Seek” may be one of her more instantly relatable works, as well, given how listeners could easily relate the song's lyricism to a past hefty romantic issue of their own."

FDRMX writer John Mychal Feraren reviewed Henry's single "Grow". He said that "the real standout in the song is the impeccable songwriting skills of Holly. She was able to establish such lyrical depth in 'Grow' that fully matches her naturally silky tone."

Music Historian blogger Patricia Trutescu interviewed Henry and wrote after the release of "Hide and Seek": "the most intriguing quality about this developing artist is that while she returns with a voice we are all familiar with, we now have the opportunity to enjoy something more refreshing – beautiful and youthful voice that expresses an old soul." Music Historian guest blogger Gary Reese wrote after the release of "*The Orchard* EP that it "successfully plays to audiences attracted to Holly's angelic voice and allegorical lyrics. The EP represents a sophisticated evolution of Holly's musical talent, combined with a remarkable collaborative achievement by producer and backup instrumentalist Matt Patrick." Reese reviewed *King Paten* in For Folk's Sake and wrote: "The eleven tracks of ‘King Paten’ spotlight Holly's rapidly maturing vocal talent, with stylistic influences clearly gained in covering over 100 unique artists on her internationally popular YouTube and SoundCloud channels. Lyrically, she coaxes us to hear how she felt in her youth, thus introducing fans to her pre-‘The Voice’ youthful innocence. It's a musical journey well worth exploring."

Ear to the Ground music reviewer Kori Rae Shearer wrote about *The Orchard* EP that Henry "has the type of tone and control that can gently interrupt conversation as heads turn her way. The understated use of her voice and subtle backing vocals are extremely effective. Holly is very skilled in the art of accentuating her message with her vocals, which speaks volumes of the control and talent she brings to the table."

## Discography

### Albums

| Title | Details | Track list |
| --- | --- | --- |
| *King Paten* | * Released: July 1, 2016 * Label: Garden Ghost | 11 tracks * 1. Powder Blue Earth * 2. Citrus * 3. Crawl * 4. In Between * 5. White Knuckles * 6. Intermission * 7. Warrior * 8. Coriander / Chamomile * 9. Joan of Arc * 10. Roswell * 11. King Paten |
| *softer*
(as Mooney) | * Released: October 25, 2019 * Label: Garden Ghost | 9 tracks * 1. hunny * 2. another phone * 3. softer * 4. phantom feeling * 5. stargate * 6. nzo * 7. fleur * 8. straps * 9. melatonin |

### Extended plays

| Title | Details | Track list |
| --- | --- | --- |
| *The Immigrant* | * Released: December 16, 2013 * Label: Garden Ghost | 5 tracks * 1. Paper Clips * 2. Sink * 3. The Immigrant * 4. The Ghost * 5. To Drift |
| *The Orchard* | * Released: August 21, 2015 * Label: Garden Ghost | 6 tracks * 1. Arbor * 2. Hotel * 3. The Orchard * 4. Skin * 5. Foolish Heart * 6. Better |

### Singles

* "Katie" (2013)
* "Secrets Spoken" (2013)
* "More Than Nothing" (2013)
* "Paper Love" (2013)
* "Hide and Seek" (2014)
* "Grow" (2014)
* "Float" (2018)

### Releases from *The Voice*

## InfoBox

| Holly Henry | |
| --- | --- |
| Birth name | Holly Mae Heinrich |
| Born | (1994-03-31) March 31, 1994
Hennepin, Minnesota, U.S. |
| Genres |  |
| Occupation(s) | Singer, songwriter |
| Instrument(s) | Guitar, ukulele, banjo, piano, organ, vocals |
| Years active | 2011–present |
| Labels | Garden Ghost Records |
|  | |
| Website | hollyhenrymusic.com |


title: Watsessing Avenue station

abstract: Watsessing Avenue station is a New Jersey Transit rail station in Bloomfield, New Jersey, along the Montclair-Boonton Line. It is located beneath the Bloomfield Police Benevolent Association meeting hall near the corner of Watsessing Avenue and Orange Street in Bloomfield. It is one of two stations on the line where the boarding platform is below ground level. The Watsessing station and the Kingsland station in Lyndhurst on the Main Line shared similar designs and were built about the same time. The current Glen Ridge, Bloomfield and Watsessing stations along the Montclair branch were all built in 1912 during a grade separation program by the Delaware, Lackawanna and Western Railroad. During New Jersey Transit's running of the line, two stations between Watsessing and Newark Broad Street were closed due to low ridership—the Roseville Avenue station in Newark, at the junction with the Morristown Line on September 16, 1984, and Ampere station in East Orange on April 7, 1991. The word "Watsessing" is a Native American term that translates to "mouth of the creek". The station has been on the New Jersey State Historic Preservation Office listings since March 25, 1998, the last of the four stations from East Orange to Glen Ridge to receive the listing. On September 14, 2005, the entire Montclair Branch was added to the same listings, although Ampere, Bloomfield and Glen Ridge stations have been on the listings since March 17, 1984.

text:
# Watsessing Avenue station

**Watsessing Avenue station** (also known as **Watsessing**) is a New Jersey Transit rail station in Bloomfield, New Jersey, along the Montclair-Boonton Line. It is located beneath the Bloomfield Police Benevolent Association meeting hall (which formerly served as the station building) near the corner of Watsessing Avenue and Orange Street in Bloomfield. It is one of two stations on the line where the boarding platform is below ground level (the Glen Ridge station, two stops away from it, is the other). The Watsessing station and the Kingsland station in Lyndhurst on the Main Line shared similar designs (both station platforms are located below street level) and were built about the same time.

The current Glen Ridge, Bloomfield and Watsessing stations along the Montclair branch were all built in 1912 during a grade separation program by the Delaware, Lackawanna and Western Railroad. During New Jersey Transit's running of the line, two stations between Watsessing and Newark Broad Street were closed due to low ridership—the Roseville Avenue station in Newark, at the junction with the Morristown Line on September 16, 1984, and Ampere station in East Orange on April 7, 1991. The word "Watsessing" is a Native American term that translates to "mouth of the creek".

The station has been on the New Jersey State Historic Preservation Office listings since March 25, 1998, the last of the four stations from East Orange to Glen Ridge to receive the listing. On September 14, 2005, the entire Montclair Branch was added to the same listings, although Ampere, Bloomfield and Glen Ridge stations have been on the listings since March 17, 1984.

## Station layout and services

Watsessing Avenue station is located on the corner of Watsessing Avenue and Orange Street at Westinghouse Plaza in Bloomfield and is just blocks from Bloomfield's borders with East Orange and Orange. The former depot is currently used by the Bloomfield Police Benevolent Association. There are two below-street-level platforms at the Watsessing station. Ticket vending machines are available at street level on Watsessing Avenue.

The station also has two parking lots for use. Maintained by the Bloomfield Parking Authority, the first is on Westinghouse Plaza (near the former Westinghouse Lamp Plant) and has fourteen parking spaces. The parking uses daily parking rules, paying six days a week at $0.25 an hour, except for Sunday, when parking is free. A second lot is available at the intersection of Myrtle Street and Walnut Street. It has forty-five parking spots, two of which are handicap accessible. The lot also contains permit spaces six days a week and free on Sundays, with a cost of $20 parking per quarter (three months).

The station has low-level side platforms that are not handicap accessible. The two nearest accessible stations are Newark Broad Street and Bay Street.

## History

### Delaware, Lackawanna and Western ownership (1856–1976)

The history of a station at Watsessing Avenue in the Watsessing district of Bloomfield dates back to the Newark and Bloomfield Railroad, established in 1856. The station, a houseless station off of Dodd Street, was deemed first as Doddtown by a railroad conductor. This name soon gave way to Watsessing, and in 1865, the line was bought by the Morris and Essex Railroad, running through trains. The Morris and Essex Railroad was soon bought out by the Delaware, Lackawanna and Western Railroad, and a new station was built, deemed Watsessing.

In 1911, as the Delaware, Lackawanna and Western Railroad continued the project of eliminating at-grade crossings between streets and railroads, the Montclair Branch was the next to receive the structural change. Bloomfield criticized the Lackawanna railroad for making a disgrace of the community. The railroad proposed using $700,000 (1911 USD) of funds to construct a brand new station at Watsessing Avenue along with a new downtown Bloomfield station. The cost of elevating and depressing the railroad came up to about $20,000 (1911 USD) for the Lackawanna. This contract by the railroad and township was approved after negotiations dating back to 1908. The negotiations included a park to be built between both stations on both sides of the railroad. The park land cost the township $50,000 to buy for the construction, and it was to be turned over to the Essex County Park Commission. The former station depot was razed in September 1910 for the construction project.

Construction was completed on a 1.5 miles (2.4 km) long segment of the Montclair Branch from East Orange to Glen Ridge and was opened to rail service on November 15, 1912. The project laid 91 pounds (41 kg) of track in addition to steel ties and stone ballast. The station has concrete crossings at Dodd Street, Arlington Avenue and Watsessing Avenue along with a new crossing of the Erie Railroad's Orange Branch just south of the station. The design of Watsessing Avenue's new station was difficult due to the limited right-of-way. While making the separation, a new trench had to be dug, which included retaining walls that prevented moving the existing track alignment to delay railroad traffic. When the station was finished, tracks were shifted to make room for a second track. The station depot was built over the railroad tracks with four concrete arches to support the building. A four-inch (10 cm) ceiling was constructed on the arches, and the station was widened to take more volume of train service.

The station served as the third station on the Montclair Branch, which was first electrified by technology created by Thomas Alva Edison in 1930. The overhead catenary wires were installed, making the line the first electrically run line on the Delaware, Lackawanna and Western alignment. The station remained in service for the Lackawanna Railroad for three more decades, when the railroad merged with the Erie Railroad on October 17, 1960. Although the now Erie-Lackawanna Railway continued to run the Montclair Branch, it reduced service, reducing the once two-rail alignment to one lone track, and removed most of the tracks at Lackawanna Terminal in Montclair. On April 1, 1976, the station was transferred to the Consolidated Rail Corporation (Conrail) as the Erie-Lackawanna Railway was dissolved into the Conrail program.

### New Jersey Transit and historical status (1979–present)

In 1979, New Jersey Transit was formed to run bus and train service in place of Conrail and the New Jersey Department of Transportation. In 1983, New Jersey Transit took over rail service from Conrail, and just one year later, the line became a temporary diesel rail line when the overhead catenary wires had to be adjusted for conversion to higher electrical voltage. The station itself was still the third on the Montclair Branch, but the line now ran into a new station at Bay Street in Montclair. On March 17, 1984 all stations on the Montclair Branch but the Roseville Avenue station in Newark, Bay Street and Watsessing Avenue were added to the State Register of Historic Places. The same would occur on June 22, 1984 at the national level with Ampere, Glen Ridge and Bloomfield Stations being added to the National Register of Historic Places. On September 16, 1984, Roseville Avenue station was closed, and just over six years later, on April 7, 1991, Ampere station in East Orange was closed. Since then, Watsessing Avenue has been the first station New Jersey Transit has served on the Montclair Branch, although East Orange has proposed reopening the station at Ampere as part of a redevelopment plan for the Ampere district.

On March 25, 1998, the station at Watsessing Avenue was given the State Historical Preservation Organization honor that Ampere, Glen Ridge and Bloomfield stations received just fourteen years prior. The station continued to receive service through the opening of Montclair Connection on September 30, 2002, which ended service as the Montclair Branch and began as the Montclair-Boonton Line, still the first station on the line after Newark Broad Street Station. On September 10, 2007, New Jersey Transit announced the canopies of the old station, then 95 years old, were to be restored and repaired. The service contracted a $1.7 million project to Watertrol Incorporated of Cranford. At that time, the station served an average of 200 people daily. Improvements for the station included brand new canopy lighting, repairs to the cantilever canopies, replacement of stairways and fencing, along with changing roof tiles and a new drainage system to replace the 1912 version. When the construction was finished in October 2008, the station now served nearly 450 people daily on average. A ceremony to mark its completion was held on October 30, 2008.

* Kingsland station – The design of Watsessing Avenue with the depot above the tracks was a replica of the style used at Kingsland.

## InfoBox

| Watsessing Ave. | |
| --- | --- |
| Watsessing Avenue station as viewed from its outbound platform, showing results of rehabilitation project undertaken in 2008. | |
| General information | |
| Location | Watsessing Avenue
Bloomfield, New Jersey |
| Coordinates | 40°46′58″N 74°11′55″W / 40.7827°N 74.1986°W / 40.7827; -74.1986 |
| Owned by | New Jersey Transit |
| Platforms | 2 side platforms |
| Tracks | 2 |
| Connections | NJ Transit Bus: 94 |
| Construction | |
| Structure type | Below-grade |
| Platform levels | 2 |
| Parking | 59 spaces, 2 accessible spaces |
| Bicycle facilities | Racks |
| Accessible | No |
| Other information | |
| Station code | 602 (Delaware, Lackawanna and Western) |
| Fare zone | 4 |
| History | |
| Rebuilt | 1912 |
| Electrified | September 3, 1930 |
| Previous names | Doddtown
Watsessing |
| Key dates | |
| September 1910 | Original station depot razed |
| Passengers | |
|  | |
| 2017 | 434 (average weekday) |
|  | |
|  | |
| Services | |
| Preceding station  NJ Transit Following station  Bloomfieldtoward Hackettstown  Montclair-Boonton Line  Newark Broad Streettoward New York Penn Station or Hoboken | |
|  | |
|  | |
| Former services   Preceding station  NJ Transit Following station  Bloomfieldtoward Bay Street  Montclair Branch  Ampere(before 1991)toward Hoboken  Preceding station Delaware, Lackawanna and Western Railroad Following station  Bloomfieldtoward Montclair  Montclair Branch  Amperetoward Hoboken | |
|  | |
|  | |
| Location | |


title: Howa

abstract: Howa Machinery, Ltd. Howa Machinery, Ltd. is a Japanese machinery manufacturer known internationally for their production of military and civilian firearms. They also manufacture products such as machine tools, sweeping vehicles and windows and doors.

text:
# Howa

**Howa Machinery, Ltd.** (豊和工業株式会社, *Hōwa Kōgyō Kabushiki-gaisha*) is a Japanese machinery manufacturer known internationally for their production of military and civilian firearms. They also manufacture products such as machine tools, sweeping vehicles and windows and doors.

## History

**Toyoda's Loom Works, Ltd.** (豊田式織機株式会社) was established by Sakichi Toyoda in February 1907. In 1941, Toyoda's Loom Works merged with **Showa Heavy Industries** (昭和重工業株式会社, established in 1936 to produce rifles, artillery shells and airplane parts) and was renamed **Howa Heavy Industries, Ltd.** (豊和重工業株式会社). The company was renamed to its current name at the end of World War II and restarted manufacturing textile machinery.

### Weapon manufacturing

#### Military use

Toyoda's Loom Works began manufacturing armaments in 1932.

##### During World War II

Since 1940, Howa has been heavily involved in the Japanese armaments industry, and was involved in manufacturing the famous Arisaka rifle series including Type 99 rifle, parts of Type 38 rifle, artillery pieces, airplane parts, and ammunition. Many of their World War II-era weapons are highly sought after collectors' items.

##### Post WWII

Howa created copies of the US M1 Garand and the M1 carbine for the Japanese Self Defense Forces during the early days of the Cold War.

Howa also designed and manufactured firearms for JGSDF use, including the following types (models):

During the early 1970s, Howa produced the AR-18 and AR-180 5.56mm assault rifle as a license from Armalite Inc. of Costa Mesa, California, which marketed the rifle to various military forces. Japanese government restrictions on the sales of military small arms to foreign countries eventually forced Howa to cease production of the AR-18/AR-180, moving production back to Armalite.

#### Civilian use

Howa has produced a long line of civilian hunting and target practice rifles in a range of calibers. Howa has also manufactured products or components for other firearm companies, such as Mossberg, Smith & Wesson, and Weatherby. The Smith & Wesson Model 1000 shotgun of the 1970s and 1980s was designed and manufactured by Howa.

* Howa M300: Based on the M1 / M2 carbine military automatic rifle designed in the United States, it was developed for hunting and sports. Produced between 1960-1996.
* Howa Golden Bear: A bolt-action rifle that was once manufactured by Howa Industries between 1967-1979.
* Howa M1500: A bolt-action rifle, is produced in sporting, varmint and heavy barrel configurations, which are sold in the U.S. via Legacy Sports International. In Canada, various retailers stock Howa rifles, including Prophet River, Cabela's, and North Pro Sports. In the United Kingdom, Highland Outdoors imports the Howa rifles.

## Sources

## InfoBox

Howa Machinery, Ltd.
|
|  |
| Native name | 豊和工業株式会社 |
| Company type | Public (K.K) |
| Traded as | TYO: 6203
NAG: 6203 |
| ISIN | JP3840600005 |
| Industry | Machinery |
| Founded | February 9, 1907 (1907-02-09) |
| Headquarters | Kiyosu city, Aichi Prefecture 452-8601, Japan |
| Key people | Takahiro Tsukamoto
(President) |
| Products |  |
| Revenue | JPY 22.3 billion (FY 2018)
(US$ 202 million) |
| Net income | JPY 1.1 billion (FY 2018)
(US$ 10 million) |
| Number of employees | 849 (consolidated, as of March 31, 2019) |
| Website | www.howa.co.jp |
| **Footnotes / references**
 | |


title: Utada the Best

abstract: Utada the Best is Japanese pop/R&B singer-songwriter Hikaru Utada's first English language compilation album, released on November 24, 2010, the same day as Utada Hikaru Single Collection Vol. 2. The album is a "best of" album featuring her more prominent and popular songs from her English career as Utada.

text:
# Utada the Best

***Utada the Best*** is Japanese pop/R&B singer-songwriter Hikaru Utada's first English language compilation album, released on November 24, 2010, the same day as *Utada Hikaru Single Collection Vol. 2*. The album is a "best of" album featuring her more prominent and popular songs from her English career as Utada.

## Background

The album was announced via her Japanese Utada site (www.Utada.jp), without much in way of promoting or officially announcing it. It has also been included in various other sources. The album is available for sale in Japan only. *Utada the Best* features both songs from her 2004 album *Exodus*, and her 2009 album *This Is the One*, as well as previously-released maxi-CD remixes by various artists. The album features both singles and non-singles from both albums.

The cover of *Utada the Best* is the same picture for that of her "Exodus '04" single cover.

Utada herself stated that the release of the compilation album is "entirely against \[her\] will". She also said "I understand that if it doesn't sell I'm the one who will take the hit, but to be honest, I don't want my fans putting down money for something that my heart isn't in." She posted on her official Twitter account: "The release of "Utada the best" is entirely against my will. I wish that my fans won't have to buy it. There's no new material in it."

## Commercial performance

The album sold a total of 20,278 copies, peaking at No. 12 on Oricon charts. This is Utada's overall lowest selling album released in Japan, and is her only album (other than Precious) that did not reach No. 1 on any chart in Japan.

## Track listing

| No. | Title | Producer(s) | Length |
| --- | --- | --- | --- |
| 1. | "Come Back to Me" (from *This Is the One*) | Utada, M.S. Eriksen, T.E. Hermansen | 3:58 |
| 2. | "Easy Breezy" (from *Exodus*) | Utada | 4:03 |
| 3. | "Merry Christmas Mr. Lawrence – FYI" (from *This Is the One*) | Utada, M.S. Eriksen, T.E. Hermansen, Ryuichi Sakamoto | 3:49 |
| 4. | "You Make Me Want to Be a Man" (from *Exodus*) | Utada | 4:37 |
| 5. | "This One (Crying Like a Child)" (from *This Is the One*) | Utada, M.S. Eriksen, T.E. Hermansen | 4:30 |
| 6. | "Exodus '04" (from *Exodus*) | Utada, Timbaland | 4:32 |
| 7. | "Apple and Cinnamon" (from *This Is the One*) | Utada, M.S. Eriksen, T.E. Hermansen | 4:39 |
| 8. | "Automatic Part II" (from *This Is the One*) | Utada, C. "Tricky" Stewart, S. Hall | 3:01 |
| 9. | "Devil Inside" (from *Exodus*) | Utada | 3:58 |
| 10. | "Kremlin Dusk" (from *Exodus*) | Utada | 5:14 |
| 11. | "Sanctuary (Opening)" (from *This Is the One* \[Bonus Track\]) | Utada | 4:25 |
| 12. | "Sanctuary (Ending)" (from *This Is the One* \[Bonus Track\]) | Utada | 5:58 |
| 13. | "Exodus '04 (JJ Flores Double J Radio Mix)" (from *Exodus '04* \[Single\]) | Utada, JJ Flores | 3:44 |
| 14. | "Devil Inside (RJD2 Remix)" (from *Devil Inside* \[Single\]) | Utada, RJD2 | 4:07 |
| 15. | "Come Back to Me (Tony Moran & Warren Rigg Radio Edit)" (from *Come Back to Me* \[Single\]) | Utada, M.S. Eriksen, T.E. Hermansen | 4:33 |
| 16. | "Dirty Desire (Mike Rizzo Radio Edit)" (from *Dirty Desire* \[Single\]) | Utada, Stewart | 3:34 |

## Release history

| Region | Date | Format |
| --- | --- | --- |
| Japan | November 24, 2010 | CD, digital download |
| | November 26, 2010 | Rental CD |

## InfoBox

| Utada the Best | |
| --- | --- |
|  | |
| Greatest hits album by Hikaru Utada | |
| Released | November 24, 2010 (2010-11-24) |
| Recorded | 2004–2009 |
| Genre |  |
| Length | 68:39 |
| Label | Universal Music Japan |
| Compiler | Universal Music Japan |
| Hikaru Utada chronology | |
| *This Is the One*
(2009) ***Utada the Best***
(2010) *Fantôme*
(2016) | |


title: Folie Saint James

abstract: The Folie St. James was a French landscape garden created between 1777 and 1780 in the Paris suburb of Neuilly-sur-Seine by Claude Baudard de Saint James, the treasurer of the French Navy under Louis XV of France. It was the work of landscape architect François-Joseph Bélanger, who had designed the garden of the Bagatelle for the Comte d'Artois. Saint James instructed Bélanger: "make what you want as long as it's expensive." The main attraction of the garden was a collection of forty-eight fabriques, or architectural constructions, placed on both sides of the avenue de Longchamp, and connected by two tunnels. The garden also featured a winding stream, crossed by numerous bridges. The fabriques included kiosques, a Chinese pavilion used an icehouse, temples, a thatched cottage, and an enormous artificial rock formation created of blocks of stone carried in carts from the Forest of Fontainebleau. The rock formation was forty three metres long, eighteen metres wide, and twelve metres high. The portico of a Doric temple was placed in an alcove of the rocks, and two staircases led up to view platforms on top of the rocks. Behind the rocks was a cascade of water which flowed into the stream. Inside the artificial rock hill was a winding tunnel encrusted with minerals and crystals, and several grottoes, as well as a bathing room with a vaulted ceiling and divans. The Scottish gardener Thomas Blaikie described the Folie this way: "This garden is, without doubt, an example of extravagance rather than taste. There is a rock built in front of the house, or rather an arch made of great blocks of stone over which water seems to flow. But although it was built at great cost, it has nothing to do with nature or natural beauty, since the rocks are mixed with carved stones, and there is a small Corinthian temple in the centre, and all the rest is equally ridiculous, because there is no mountain or height in the vicinity to form this enormous mass of rocks." Saint James was ruined by the financial crisis of 1780, and in 1787 all his belongings, including the Folie, were seized and sold. The garden was partially destroyed in 1895.

text:
# Folie Saint James

The **Folie St. James** was a French landscape garden created between 1777 and 1780 in the Paris suburb of Neuilly-sur-Seine by Claude Baudard de Saint James, the treasurer of the French Navy under Louis XV of France. It was the work of landscape architect François-Joseph Bélanger, who had designed the garden of the Bagatelle for the Comte d'Artois. Saint James instructed Bélanger: "make what you want as long as it's expensive."

The main attraction of the garden was a collection of forty-eight fabriques, or architectural constructions, placed on both sides of the avenue de Longchamp, and connected by two tunnels. The garden also featured a winding stream, crossed by numerous bridges. The fabriques included kiosques, a Chinese pavilion used an icehouse, temples, a thatched cottage, and an enormous artificial rock formation created of blocks of stone carried in carts from the Forest of Fontainebleau. The rock formation was forty three metres long, eighteen metres wide, and twelve metres high. The portico of a Doric temple was placed in an alcove of the rocks, and two staircases led up to view platforms on top of the rocks. Behind the rocks was a cascade of water which flowed into the stream. Inside the artificial rock hill was a winding tunnel encrusted with minerals and crystals, and several grottoes, as well as a bathing room with a vaulted ceiling and divans.

The Scottish gardener Thomas Blaikie described the Folie this way: "This garden is, without doubt, an example of extravagance rather than taste. There is a rock built in front of the house, or rather an arch made of great blocks of stone over which water seems to flow. But although it was built at great cost, it has nothing to do with nature or natural beauty, since the rocks are mixed with carved stones, and there is a small Corinthian temple in the centre, and all the rest is equally ridiculous, because there is no mountain or height in the vicinity to form this enormous mass of rocks."

Saint James was ruined by the financial crisis of 1780, and in 1787 all his belongings, including the Folie, were seized and sold. The garden was partially destroyed in 1895.

48°52′56″N 2°15′31″E / 48.88222°N 2.25861°E / 48.88222; 2.25861


title: Lubefu

abstract: Lubefu is a community in Sankuru province of the Democratic Republic of the Congo. It is the administrative center of the Lubefu territory. Lubefu was established as a government station by the Belgian colonial administration, lying on the Lubefu River about 100 miles (160 km) upstream from Bena Dibele, a town on the Sankuru just below the point where it is joined by the Lubefu.

text:
# Lubefu

**Lubefu** is a community in Sankuru province of the Democratic Republic of the Congo. It is the administrative center of the Lubefu territory.

Lubefu was established as a government station by the Belgian colonial administration, lying on the Lubefu River about 100 miles (160 km) upstream from Bena Dibele, a town on the Sankuru just below the point where it is joined by the Lubefu.

## InfoBox

| Lubefu | |
| --- | --- |
| Commune | |
| Lubefu | |
| Coordinates: 4°43′00″S 24°25′00″E / 4.71667°S 24.41667°E / -4.71667; 24.41667 | |
| Country | DR Congo |
| Province | Sankuru |
| Territory | Lubefu |
| Time zone | UTC+2 (Central Africa Time) |


title: Hog Island (San Joaquin County)

abstract: Hog Island is an island in the San Joaquin River, and is one of many islands which constitute the Sacramento–San Joaquin River Delta. It was used for agriculture in the early 20th century, but has now mostly become marsh or submerged land; it remains a spot for fishing, particularly channel and blue catfish.

text:
# Hog Island (San Joaquin County)

**Hog Island** is an island in the San Joaquin River, and is one of many islands which constitute the Sacramento–San Joaquin River Delta. It was used for agriculture in the early 20th century, but has now mostly become marsh or submerged land; it remains a spot for fishing, particularly channel and blue catfish.

## Geography

Hog Island is located in San Joaquin County, California, and is not managed by any reclamation district. Hog Island was separated from Spud Island to its southwest by the dredging of the Hog Island Cut; the two islands are in the San Joaquin River, at a point where McDonald Island lies to the west of the river and Rindge Tract to its east. The United States Geological Survey (USGS) gives its elevation as 13 ft (4.0 m).

## History

The existence of a "Hog Island" in the Sacramento–San Joaquin River Delta is difficult to determine, since there are multiple islands with the same name. Even within northern California, there exists a Hog Island approximately 100 mi (160 km) northwest in Tomales Bay (referenced by that name as early as 1916) as well as a Hog Island 60 mi (97 km) northwest in the Petaluma River (referenced as early as 1914). Nevertheless, references to a Hog Island in the Delta area can be found as early as 1880, and it first appears labeled on a 1910 USGS map surveyed in 1907 to 1908. However, the orientation of the islands was different on the 1910 map, in which they are separated by the Deadman Reach segment of the San Joaquin River (running north to south).

In the early 20th century, the island was used for agriculture; references to crops being grown on Hog Island appear in 1914, and a 1923 survey by the California Department of Water Rights gave its area as 100 acres (40 ha), with 80 acres (32 ha) irrigated for crops, all of which was devoted to the cultivation of wheat.

In June 1931, Hog and Spud Islands were mentioned in an engineering report on dredging in the Delta; the Hog Island Cut was being dredged to separate them, and levees were in the process of being constructed on both sides of the new cut with clamshell dredgers. From then onward, Hog Island appears on maps in its current shape. Over time, it began to subside; a 1939 USGS map shows it with a small slough in its interior on the northwest side, and a 1952 map shows it with substantial portions of its northern half as marshland. In 1974, a report from the United States Army Corps of Engineers noted that dredging spoils had been deposited on the island in the past, and that approximately 15 acres (6.1 ha) of the island's southern end was grassland (with the remainder of it being either marsh or submerged). This agrees with the state of the island as shown in a 1978 USGS map.

The owners of Hog and Spud Islands spent many years attempting to open them to the public for recreational use; while plans were approved by the San Joaquin County Board of Supervisors in 2007, additional agencies required additional documents. In June 2011, the *Modesto Bee* reported that plans to build a family science camp were "moving forward with the blessing of the Army Corps of Engineers", and that new docks would be constructed later that summer.

Hog Island is known as a fishing spot in the Delta, where halibut and striped bass have been caught.

## InfoBox

Hog Island
| USGS aerial imagery of Hog Island | |
| --- | --- |
| Hog IslandHog IslandHog Island | |
| Geography | |
| Location | Northern California |
| Coordinates | 38°00′15″N 121°26′53″W / 38.00417°N 121.44806°W / 38.00417; -121.44806 (Hog Island) |
| Adjacent to | Sacramento–San Joaquin River Delta |
| Highest elevation | 13 ft (4 m) |
| Administration | |
| **United States** | |
| State | California |
| County | San Joaquin |


title: Watersheds of North America

abstract: Watersheds of North America are large drainage basins which drain to separate oceans, seas, gulfs, or endorheic basins. There are six generally recognized hydrological continental divides which divide the continent into seven principal drainage basins spanning three oceans and one endorheic basin. The basins are the Atlantic Seaboard basin, the Gulf of Mexico basin, the Great Lakes-St. Lawrence basin, the Pacific basin, the Arctic basin, the Hudson Bay basin, and the Great Basin. Together, the principal basins span the continent with the exception of numerous smaller endorheic basins. The Atlantic Seaboard basin in eastern North America drains to the Atlantic Ocean; the Great Lakes-St. Lawrence basin in central and eastern North America drains to the Gulf of St. Lawrence on the Atlantic Ocean or to the Labrador Sea; the Gulf of Mexico basin in the southern United States drains to the Gulf of Mexico, a basin of the Atlantic Ocean; the Pacific basin in western North America drains to the Pacific Ocean or the Gulf of California on the Pacific Ocean; the Arctic basin in northwestern North America drains to the Beaufort Sea of the Arctic Ocean or directly to the Arctic Ocean; the Hudson Bay basin in central northern North America drains to Hudson Bay on the Labrador Sea, or to the Arctic Sea via Foxe Basin and Fury and Hecla Strait; the Great Basin in western United States is an endorheic basin which does not drain to any ocean. The Atlantic Seaboard basin is bounded by the Atlantic Ocean to the east, the Eastern Continental Divide to the west, the Great Lakes-St. Lawrence Divide to the north, and the Lake Okeechobee endorheic basin to the south. The Gulf of Mexico basin is bounded by the Gulf of Mexico to the south, the Eastern Continental Divide to the east, the Great Lakes-St. Lawrence Divide to the northeast, the Laurentian Divide to the North, and the Continental Divide to the west. The Pacific Basin is bounded by the Continental divide to the east and Pacific Ocean to the west; the basin excludes the endorheic Great Basin in the west. The Great Basin has a closed loop boundary encompassing substantially all of Nevada, the western half of Utah and parts of Oregon, California, Idaho, and Wyoming.

text:
# Watersheds of North America

**Watersheds of North America** are large drainage basins which drain to separate oceans, seas, gulfs, or endorheic basins. There are six generally recognized hydrological continental divides which divide the continent into seven principal drainage basins spanning three oceans (Arctic, Atlantic and Pacific) and one endorheic basin. The basins are the Atlantic Seaboard basin, the Gulf of Mexico basin, the Great Lakes-St. Lawrence basin, the Pacific basin, the Arctic basin, the Hudson Bay basin, and the Great Basin. Together, the principal basins span the continent with the exception of numerous smaller endorheic basins.

The Atlantic Seaboard basin in eastern North America drains to the Atlantic Ocean; the Great Lakes-St. Lawrence basin in central and eastern North America drains to the Gulf of St. Lawrence on the Atlantic Ocean or to the Labrador Sea; the Gulf of Mexico basin in the southern United States drains to the Gulf of Mexico, a basin of the Atlantic Ocean; the Pacific basin in western North America drains to the Pacific Ocean or the Gulf of California on the Pacific Ocean; the Arctic basin in northwestern North America drains to the Beaufort Sea of the Arctic Ocean or directly to the Arctic Ocean; the Hudson Bay basin in central northern North America drains to Hudson Bay on the Labrador Sea (which may be considered part of either the Arctic Ocean or Atlantic Ocean), or to the Arctic Sea via Foxe Basin and Fury and Hecla Strait; the Great Basin in western United States is an endorheic basin which does not drain to any ocean.

The Atlantic Seaboard basin is bounded by the Atlantic Ocean to the east, the Eastern Continental Divide to the west, the Great Lakes-St. Lawrence Divide to the north, and the Lake Okeechobee endorheic basin to the south. The Gulf of Mexico basin is bounded by the Gulf of Mexico to the south, the Eastern Continental Divide to the east, the Great Lakes-St. Lawrence Divide to the northeast, the Laurentian Divide to the North, and the Continental Divide to the west. The Pacific Basin is bounded by the Continental divide to the east and Pacific Ocean to the west; the basin excludes the endorheic Great Basin in the west. The Great Basin has a closed loop boundary encompassing substantially all of Nevada, the western half of Utah and parts of Oregon, California, Idaho, and Wyoming.

## Endorheic basins

Most lakes are not actually endorheic, but endorheic basins may not have standing water, or have water only seasonally. The most significant endorheic basins are these:

## Boundary landforms

The western boundary (Eastern Continental Divide) of the Atlantic Seaboard basin and eastern boundary of the Gulf of Mexico basin is formed by the Appalachian Mountains to the North, the Piedmont Plateau and lowland ridges of the Atlantic Coastal plain to the south. The eastern boundary of the Pacific basin and western boundary of the Gulf of Mexico basin (Continental Divide) is the Rocky Mountains.

## Principal rivers of basins

A principal river of a basin is a river that drains directly to the incident ocean, sea or gulf, or into an endorheic basin.

### Great Lakes-St. Lawrence basin

### Atlantic Seaboard basin

### Gulf of Mexico basin

### Arctic basin

### Hudson Bay basin

### Pacific basin

### Great Basin

* Canadian Encyclopedia, "Drainage Basin"


title: European Technical Center

abstract: The European Technical Center (ETC) is a U.S. National Security Agency (NSA) signals intelligence facility in Mainz-Kastel, Wiesbaden, Germany. Located in Building 4009 of the U.S. Army's Mainz-Kastel Storage Station, the facility serves as the NSA's "primary communications hub" in Europe. The center was known by the code name "GODLIKELESION".

text:
# European Technical Center

The **European Technical Center** (**ETC**) is a U.S. National Security Agency (NSA) signals intelligence facility in Mainz-Kastel, Wiesbaden, Germany. Located in Building 4009 of the U.S. Army's Mainz-Kastel Storage Station, the facility serves as the NSA's "primary communications hub" in Europe. The center was known by the code name "GODLIKELESION".

## History

In December 2005, a metal object discovered near the facility was suspected of being an undetonated World War II explosive. On January 24, 2006, NSA analysts evacuated the ETC while the object was investigated, shutting down the center for a full day. The suspicious ordnance proved to be merely a "pile of junk".

In 2010, the center began experiencing technical difficulties, losing power over 150 times over the period of a few months. These events prompted the United States to expand and refurbish the facility, which was completed on September 19, 2011. Since then, the center has served as the NSA's "primary communications hub" in Europe, intercepting and forwarding large amounts of data to the NSA, U.S. military, and U.S. strategic partners in Europe, Africa and the Middle East.

The facility was disclosed in 2014 in documents released by the former NSA contractor Edward Snowden.

50°00′41″N 8°17′16″E / 50.0114°N 8.2877°E / 50.0114; 8.2877


title: Norrin

abstract: Norrin, also known as Norrie disease protein or X-linked exudative vitreoretinopathy 2 protein (EVR2) is a protein that in humans is encoded by the NDP gene. Mutations in the NDP gene are associated with the Norrie disease.

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# Norrin

**Norrin**, also known as **Norrie disease protein** or **X-linked exudative vitreoretinopathy 2 protein** (EVR2) is a protein that in humans is encoded by the *NDP* gene. Mutations in the NDP gene are associated with the Norrie disease.

## Function

Signaling induced by the protein Norrin regulates vascular development of vertebrate retina and controls important blood vessels in the ear. Norrin binds with high affinity to Frizzled 4, and Frizzled 4 knockout mice exhibit abnormal vascular development of the retina.

## Clinical significance

NDP is the genetic locus identified as harboring mutations that result in Norrie disease. Norrie disease is a rare genetic disorder characterized by bilateral congenital blindness that is caused by a vascularized mass behind each lens due to a maldeveloped retina (pseudoglioma).

## Further reading

## InfoBox

| NDP | | | |
| --- | --- | --- | --- |
|  | | | |
| Available structures  PDB Ortholog search: PDBe RCSB | | | |
| Identifiers | | | |
| Aliases | NDP, EVR2, FEVR, ND, Norrie disease (pseudoglioma), norrin cystine knot growth factor, norrin cystine knot growth factor NDP | | |
| External IDs | OMIM: 300658; MGI: 102570; HomoloGene: 225; GeneCards: NDP; OMA:NDP - orthologs | | |
|  | | | |
|  | | | |
|  | | | |
|  | | | |
| Orthologs  Species **Human** **Mouse**  Entrez   Ensembl   UniProt   RefSeq (mRNA)     NM\_000266   NM\_010883   RefSeq (protein)     NP\_000257   NP\_035013   Location (UCSC) Chr X: 43.95 – 43.97 Mb Chr X: 16.75 – 16.78 Mb  PubMed search | | | |
| Wikidata | | | |
| View/Edit Human View/Edit Mouse | | | |


title: Lefourba-Foulbé

abstract: Lefourba-Foulbé is a village in the Sabce Department of Bam Province in northern-central Burkina Faso. It has a population of 63.

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# Lefourba-Foulbé

**Lefourba-Foulbé** is a village in the Sabce Department of Bam Province in northern-central Burkina Faso. It has a population of 63.

## InfoBox

| Lefourba-Foulbé | |
| --- | --- |
| Country | Burkina Faso |
| Region | Centre-Nord Region |
| Province | Bam Province |
| Department | Sabce Department |
| Population (2005) | |
| Total | 63 |
| Time zone | +2 |