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Results tagged with riemannian-geometry
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user 16750
Riemannian Geometry is a subfield of Differential Geometry, which specifically studies "Riemannian Manifolds", manifolds with "Riemannian Metrics", which means that they are equipped with continuous inner products.
5
votes
2
answers
902
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Example for Busemann function is not an exhaustion when Ricci $\ge 0$
For an open complete Riemannian manifold $M$ with non-negative sectional curvature, the Busemann function defined below is a convex exhaustion function (by Cheeger-Gromoll's proof of soul theorem)
Th …
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2
votes
1
answer
481
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Minimum set of subharmonic function in $\mathbb R^n$
Let $f :\mathbb R^n\to \mathbb [0, \infty)$ be a (continuous, $C^2$, or smooth) subharmonic function with minimum value $0$. Then we know the sublevel set $f^{-1}((-\infty, c])$ is mean convex for $c …
- 891
19
votes
2
answers
2k
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Area of distance sphere in manifold with Ricci $\ge 0$.
Let $M$ be a open complete manifold with Ricci curvature $\ge 0$.
By a theorem of Calabi and Yau, the volume growth of $M$ is at least of linear.
I am wondering whether the following statement is true …
- 891
5
votes
1
answer
288
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Dimension of certain subgroup of isometry group of positively curved manifold
Let $M$ be a closed $n$-dimensional Riemannian manifold with positive sectional curvature.
Let $G$ be a close subgroup of isometry group ${\rm Iso}(M)$. Suppose the action of $G$ on $M$ is not transit …
- 891