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103 questions
0
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volume of a submanifold implies bounds on curvature
I would like to ask the following question: Suppose an m-dimensional manifold in an n-dimensional euclidean space, choose some point on this manifold and take an n-dimensional ball of radius R centred ...
- 2,481
6
votes
0
answers
269
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Negative curvature in the middle of $R^{3}$
What's a simple example of metric on $R^{3}$ which has negative scalar curvature inside of a (limited) set N, and is equal to the standard (Euclidean) metric outside?
Basically, I am asking for a ...
- 61
3
votes
1
answer
583
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geometric meaning of Ricci-flatness
What is the geometric meaning of Ricci-flatness? We know that if the Riemann tensor at a point vanished, manifold is flat at this point. but I don't know When the Ricci tensor vanished at a point, ...
- 25.3k