Tangent Vector Spaces: Clarifying Dimension and Interpretation

[rtharbaugh1]

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 k-dimensional 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 k-1? I am imagining the tangent vector space at a point on a three-sphere as a 2-D 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

 

[matt grime]

The surface of the Earth is 2-d (locally). That it lives ina 3d space is neither here nor there.

I don't know about anyone else, but my definition of a k-dimensional manifold is that locally (i.e. the tangent space) a k-dimensional vector space. So of course it should not be k-1. Unless you think that the surface of the Earth is 1 dimensional.

 

[daveb]

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

 

[HallsofIvy]

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 k-dimensional 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 k-1? I am imagining the tangent vector space at a point on a three-sphere as a 2-D 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

As others have pointed out, the 2-D disk you are seeing as the tangent space is the tangent space to the 2-dimensional surface of the sphere, not to the 3- dimensional sphere itself.

 

[jbusc]

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

 

[mathwonk]

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.

 

[Data]

As others have pointed out, the 2-D disk you are seeing as the tangent space is the tangent space to the 2-dimensional surface of the sphere, not to the 3- dimensional sphere itself.

And to speak precisely, sphere always means just the surface. If you want the volume contained in it, that's a ball!

 

[rtharbaugh1]

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