What dimensions do symplectic manifolds occour in?
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every symplectic manifold M is of even dimension 2n
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Why are you asking this again?  I just answered you.\n\nAgain:\nThere are "levels" to dimensioning.\n\nThe major dimensions are the overall height, width, depth of the whole object. \n\nThe next level down will sub-divide that major dimension to describe sub-components.\n\nAs example: Consider a house plan. \n* From one extreme side to the other extreme side is a MAJOR dimension (the overall width). \n* In between those extreme outer wall edges are interior walls that need to be located as well. So the next layer of dimensions (MINOR) still goes from one side to the other but pulls lots of dimensions from lots of walls. \n* Also, there are porches, corners for inset windows, etc Another MINOR level picks up those points.\n* And so on.\n\nIn your 3D drawing: \n* you only have 6 MAJOR dimensions to a cube shape anyway. Depending on how you rotate the drawing object, you see 3 of them and 3 will be hidden (back side).
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Also depends on your scale. At microscopic scales, a strings length is fractal, and thereby infinite.\n\nJust think of measuring a coastline length at a microscopic scale.
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What do you mean by dimensions? I don't understand the question you are asking.\n\nFor information you can visit here http://www.himalayanacademy.com/resources/books/hbh/hbh_ch-6.html\n\nIt is a good site about Hinduism as well as http://www.himalayanacademy.com
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log into www.wikipedia.org and search for lie group...there is a detailed description and all the formulas as well and its free.
858799
This depends on the specific Lie group. The exponential map for GL^+(n,R) is not surjective (for n=2, [[-1 0][0 -2]] is not an exponential), but is is surjective for GL(n,C). Notice that it cannot be surjective unless the group is connected [GL(n,R) is not]. The map tends not to be injective. For example exp[[0 0][0 0]] and exp[[2pi*i 0][0 0]] are both the identity. Compactness considerations give negative answers for bijectivity for all compact Lie groups.\n\nThe exponential map is bijective on a neighborhood of the identity for every Lie group. The reason is that the differential map is essentially the identity and so the Inverse Function Theorem applies. Cleartly it is both injective and surjective for R^n.
1141293
1-4 are 3 spacial and 1 time\n\n5-11 are the what string theory are trying to prove, being really small dimesions curled up into little multidimensional balls.. \nthese dimensions would explain things like electromagnetism and gravity
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http://mathworld.wolfram.com/PoincareConjecture.html
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I fear i had even thought about the ∞ dimension once. Nearly anything could be described as rotations there. It has took a long time to get a foot on earth again.\n\nTime is not the real 4th dimension. Perhaps c*t. Look at this 'Minkowski space' thingy:\n\nhttp://en.wikipedia.org/wiki/Minkowski_space\n\nIf you measure across space AND time, there are negative distances. What shoud a negative distance mean in reality?\nIf you consider time as the 4th dimension, it is just a convenient way to write down long terms shortly or to visualize complex relations concisely. It isn't more than just mathematics.\n\n\n"As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality." \n\nAlbert Einstein (personally himself)
83422
Scientist belive there are several dimesions.  Remember dimensions are based on measurement.  Length width height, 3 dimesions so we are in the third dimension.  A showdow for example only has length and width so it is the 2nd dimseion.  We cannot comprehend other demensions because we live in a 3d world where only LWH exist.  It is like asking your shadow what is tall, he would know because he lives in the 2nd demsiond.  So there are other dimensions yes and we can dicipher them though the wonders of mathmatics and physics.\n\nGravity works likes a bowling ball on a bed.  Think of the bed as the fabric of space.  If you put a tennis ball on the bed it would "gravitate" towards the bowling ball.  This is the theory of realtivity.  A worm hole forms when a really heavy mass punches through that space fabric.  If you had a million pound bowling ball it would go right though the bed.  The same things happens in space that heavy mass punches right though to the other side forming a worm hole.
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I think u could visit this website for the general relativity:\n\nhttp://archive.ncsa.uiuc.edu/Cyberia/NumRel/GenRelativity.html
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http://en.wikipedia.org/wiki/Inner_Space
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http://www.pd.org/topos/perforations/perf7/koresh/nash-koresh.html
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I like the joke!  To play along, I define:\nr = metric r  and \nd = metric d\n\nV = (pi/3)*(d^2)*(3r - d)\n\nThis will work for the tank being any fraction of a sphere, not just 1/2
410020
Xcuse me dude, plz be clear abt wt u wanna ask abt the PRINCILES OF GEOMETeRY
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Time And Relative Dimension (or Dimensions) In Space
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Well it depends on what you mean by dimensions. Some of the "dimensions" you list are parameters or variables. \n\nHowever, according to the standard definition of dimensions, we know about 4 dimensions (3 space and one time). There have been many attempts to unify the 4 fundamental forces of matter ("unified field theory") that have postulated the existence of more dimensions. This possibly started with the Kaluza-Klein theory, creating a 5th dimension. \n\nModern string theories and other theories require more dimensions. Many different types of string theory exist, and many different suggestions for the number of dimenions exist as well in string theory. However, no one really knows which if any of these theories are correct. The current leading string theory, M-brane theory, requires 11 dimensions.