5
votes
1answer
915 views

Hodge Theory (Voisin)

I have a strong understanding of Representation Theory but am interested in learning from Voisin, Hodge Theory and Complex Algebraic Geometry. What are the prerequisites to learning from this ...
5
votes
3answers
872 views

group of diffeomorphisms of a manifold

How much has been the group of diffeomorphisms of a manifold " been studied. I got this information from wiki. " Quite a lot is known about the group of diffeomorphisms of the circle. Its Lie algebra ...
1
vote
0answers
265 views

applications of gauss-bonnet theorem

In wikipedia,I was pretty amazed to find a proof of fundamental theorem of algebra using gauss bonnet theorem.I think given how central it is to mathematics with its far reaching generalizations like ...
14
votes
1answer
621 views

What are “good” examples of string manifolds?

Based on this mathoverlow question, I would like to have a similar list for the case of string manifolds. An $n$-dim. Riemannian manifold $M$ is said to be string, if the classifying map of its bundle ...
23
votes
4answers
2k views

What are “good” examples of spin manifolds?

I'm trying to get a grasp on what it means for a manifold to be spin. My question is, roughly: What are some "good" (in the sense of illustrating the concept) examples of manifolds which are spin ...
15
votes
9answers
4k views

Open Questions in Riemannian Geometry

What are some major open problems in Riemannian Geometry? I tried googling it, but couldn't find any resources.
49
votes
13answers
5k views

What makes four dimensions special?

Do you know properties which distinguish four-dimensional spaces among the others? What makes four-dimensional topological manifolds special? What makes four-dimensional differentiable manifolds ...
5
votes
2answers
2k views

Constant curvature manifolds

In two different books I found these two related statements. The book by Jost defines a ``locally symmetric space" as one for which the curvature tensor is constant and which is geodesically ...
13
votes
3answers
2k views

A list of machineries for computing cohomology

This is not a question, but I just hope to hear more from everyone here on it. A list of ready-to-use machineries to compute the de Rham / Cech cohomology of a manifold / variety. As far as I know, I ...
17
votes
10answers
5k views

Riemannian Geometry Introductory Text

I have studied differential geometry, and am looking for basic introductory texts on Riemmanian geometry. My target is eventually Kähler geometry, but certain topics like geodesics, curvature, ...
26
votes
5answers
5k views

Doing geometry using Feynman Path Integral?

I have often heard in the folk-lore that Feynman Path Integral can be used to compute geometric invariants of a space. Coming from a background of studying Quantum Field Theory from the books like ...