- #1
rtharbaugh1
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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 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
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