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Results tagged with differential-topology
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user 125275
The study of differentiable manifolds and differentiable maps. One fundamental problem is that of classifying manifolds up to diffeomorphism. Differential topology is what Poincaré understood as topology or “analysis situs”.
13
votes
Philosophy behind the Ricci flow
The Ricci flow behaves like a heat equation for the curvature with some additional reaction terms. Intuitively, the diffusion term acts to spread the curvature out and make the geometry more homogeneo …
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3
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Arbitrary sectional curvatures at a point
To your second question, the answer is no. Using polarization, t)he sectional curvature determines the Riemann curvature tensor so if we know the sectional curvature on "enough" two planes, it is uniq …
- 6,001
18
votes
Accepted
Intuition behind manifolds which are homeomorphic but not diffeomorphic
Milnor’s original construction of an exotic $\mathbb{S}^7$ is fairly explicit and it’s reasonable to think that the resulting space is too weirdly twisted to actually be a sphere in the usual sense. S …
- 6,001
20
votes
Books in advanced differential topology
Differential Manifolds by Antoni Kosinski covers many of these topics. It also includes a very well-written proof of the h-cobordism theorem, which I found really illuminating.
- 6,001
4
votes
The purpose of connections in differential geometry
This might be my own bias, but I think differential geometry is a really natural area to study. When we look out at the world around us, we see lots of objects that seem smooth, but are not flat. As s …
10
votes
Examples of Banach manifolds with function spaces as tangent spaces
This doesn't exactly fit your criteria as it's not a Banach manifold, but one good example of an infinite dimensional space is the space of probability measures with finite second moment along with t …
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