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 ...

- 99.3k

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) ...

- 99.3k

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

Only top scored, non community-wiki answers of a minimum length are eligible

#### Related Tags

riemannian-geometry × 2648dg.differential-geometry × 1702

mg.metric-geometry × 340

reference-request × 297

ap.analysis-of-pdes × 190

smooth-manifolds × 142

curvature × 132

lie-groups × 128

complex-geometry × 127

differential-topology × 109

gt.geometric-topology × 107

fa.functional-analysis × 94

hyperbolic-geometry × 83

geodesics × 81

ds.dynamical-systems × 78

elliptic-pde × 71

sp.spectral-theory × 64

alexandrov-geometry × 64

ricci-flow × 58

at.algebraic-topology × 56

sg.symplectic-geometry × 56

conformal-geometry × 56

riemann-surfaces × 55

ag.algebraic-geometry × 53

connections × 53