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5 votes
1 answer
495 views

Volume comparison on Grassmannian

Let $G(r,n-r)$ be the Grassmannian, which can be identified with the space of all rank $r$ projection matrices in $\mathbb{R}^n$. Let $\mu$ be the uniform measure on $G(r,n-r)$. For any $\lambda_1,......
neverevernever's user avatar
4 votes
1 answer
197 views

Distance comparison in submanifold versus in the underlying manifold

Let $(M,g)$ be the (underlying) manifold, $(S,g|)$ be a submanifold. Let $a,b,c \in S$. It's not in general true that $d_M(a,b)\leq d_M(a,c) \implies d_S(a,b)\leq d_S(a,c)$. QUESTION I: The above ...
Learning math's user avatar
3 votes
0 answers
158 views

Conformal Killing vector fields on manifolds that are not asymptotically flat

Let $M = [1,\infty) \times S^2$. Equip $M$ with the metric $g = dr^2 + r^2 (\gamma + h)$ where $\gamma$ is a metric on $S^2$ and $h$ is a $(0,2)$ tensor on $M$ that satisfies $$h = O(1/r),\quad \...
Laithy's user avatar
  • 969
1 vote
0 answers
56 views

Good orbifold and Ricci flow with Dirichlet boundary conditions on $\Sigma$

An orbifold $\mathcal O$ is a metrizable topological space equipped with an atlas modeled on $\Bbb R^n/\Gamma, \Gamma<O(n)$ finite. Let $\Sigma$ be the singular locus i.e. points modeled on $\...
John McManus's user avatar