5 votes
Accepted

Does the isometry group determine the Riemannian metric?

I think that you are missing some hypotheses on the action of $G$, otherwise there are trivial counterexamples. For example, if $G\subset\mathrm{Iso}(M,g)$ is the trivial group, then one clearly ...
4 votes

Volume of submanifold as integral of delta-function

One should distinguish between the volume of the submanifold (a number that might be infinite) and the volume form, an exterior differential form $\omega$ of degree $n{-}m$ on the (presumed regular) ...
2 votes

Complex quadric as a symmetric space

So while the projective quadric is indeed a homogeneous space of the form $G/P$ for $P$ parabolic in $G$ (marking it as a generalised flag manifold) this is not the realisation of it as a symmetric ...
  • 637
2 votes
Accepted

Pair of laminations that fill on a closed surface

The answer is "no". For consider the case where $L_1 = L_2$ is a single simple closed geodesic. Now you've added the hypothesis that $L_1$ and $L_2$ have no common leaf, the answer becomes ...
  • 20.4k
2 votes
Accepted

Ricci scalar of submanifold of $\mathbf R^n$

This is an extended comment answering Ricci scalar of sub-manifold of $\mathbf R^n$ Assuming the $f_i$ are independent, at every point $x$ their gradients span the orthogonal complement to the tangent ...
  • 31.9k
2 votes

On properties of Besse spheres

It so happens I recently read this same passage in Besse's book. I am no expert, but here is how I understand it. Question 1. Let $p,q \in \mathbf{S}^2$ be two arbitrary points, and $\eta: [0,L] \to \...
  • 2,949
1 vote
Accepted

$(1+\epsilon)$-bilipschitz parametrization of Lipschitz manifold

The answer to my question is no, and a counter-example is provided by the staircase function of the fat-Cantor set, which is a Lipschitz function $f:[0,1]\to\mathbb{R}$ with the property that $\{f'=0\}...
  • 632
1 vote

A question on convexity and conjugate points

Here is a proof that works, I think—although it's probably a bit clumsier than necessary. The short version is that the geodesics in $M \setminus \partial M$ are minimizing. This means we can't ...
  • 2,949

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