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Tangent vector space question 
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#1
Sep1306, 10:48 AM

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I see in my notes (I don't carry The Encyclopedia Britannica around with me) that George Mostow, in his artical on analytic topology, says "The set of all tangent vectors at m of a kdimensional manifold constitutes a linear or vector space of which k is the dimension (k real)." Well ok, maybe it is more a paraphrase than a quote.
Shouldn't the dimension of the tangent vector space be k1? I am imagining the tangent vector space at a point on a threesphere as a 2D disk originating at the point, rather as if I had tacked a CD onto my globe of the Earth. Then on the real Earth, I am at a point, and my tangent space would be the space between me and the horizon? Say I am at sea far from any coast. Should I rather think of the tangent space as the 2d surface of the ocean, or as the 3d space in which the ocean waves occur? Thanks, R 


#2
Sep1306, 12:57 PM

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The surface of the earth is 2d (locally). That it lives ina 3d space is neither here nor there.
I don't know about anyone else, but my definition of a kdimensional manifold is that locally (i.e. the tangent space) a kdimensional vector space. So of course it should not be k1. Unless you think that the surface of the earth is 1 dimensional. 


#3
Sep1306, 01:03 PM

P: 925

a point on a 3sphere can be thought of as a point on the unit 3d sphere or the 2d unit shell. The tangent space to the unit shell is the 2d plane that is tangent at that point. But for the 3d sphere, the tangent space is 3 dimensional.



#4
Sep1306, 05:54 PM

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Tangent vector space question



#5
Sep1406, 12:24 PM

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All a kdimensional manifold is, is a space which locally "looks" like euclidean kspace. So in sufficiently "small" regions you would expect vectors to behave like they would in euclidean kspace, meaning the vectors "live" in a kdimensional space. When you consider the whole manifold again, those kdimensional spaces appear as the tangent spaces since they change as you move along the manifold.



#6
Sep1406, 06:12 PM

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a k manifold is something locally homeomorphic (or diffeomorphic) to R^k, while a k vector space is something linearly isomorphic to R^k.
the tgangent space is the linear space that best approxiamtes the manifold. It makes sense it should have the same dimension. a sphere in R^3 is locally diffeomorphic to the plane, via stereographic projection, hence a sphere is 2 dimensional. 


#7
Sep1406, 08:39 PM

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#8
Sep1506, 10:34 AM

P: 310

Thanks to all. I think I get it now. R.



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