11
votes

Accepted

### Analogous results in geometric group theory and Riemannian geometry?

I think Cheeger's inequality is a good example.
Riemannian geometry version
Let $M$ be a closed Riemannian $n$-manifold. Say that a $n-1$ dimensional submanifold $N$ separates $M$ if the complement of ...

10
votes

### Analogous results in geometric group theory and Riemannian geometry?

Here is a very classical example. As stated in the comments, Gromov was an early proponent of importing ideas from geometry to group theory, but already thirty years earlier there was work in this ...

5
votes

Accepted

### Is the vector field associated with an element of the boundary at infinity on a Hadamard manifold smooth?

This vector field is the negative of the gradient vector field of the/a Busemann function $b_u$ associated with $u$. (Below, I am assuming that the Riemannian metric is $C^\infty$.) The function is ...

4
votes

Accepted

### Lee-Parker Yamabe problem proposition 4.6

You are forgetting that, as part of the renormalization, the functions $\psi_s$ all satisfy $\psi_s(-P)=1$ (here $-P$ is the south pole). In particular, since $\psi$ is $C^2$ away from the north pole,...

4
votes

Accepted

### Convergence of spectrum

$C^0$-convergence is sufficinent.
Note that $\lambda_i$ can be defined as the least lower bound on numbers $\lambda$ such that the following property holds:
There is an $i$-dimensional subspace $W$ ...

4
votes

Accepted

### Why is this subset associated to a $2$-tensor dense?

I claim that the function $E_S$ is lower semi-continuous, meaning that for any sequence $x_k \in M$ converging to some $x \in M$, $\lim_{k \to \infty} E_S(x_k) \geq E_S(x)$.
We want to show that the ...

3
votes

### What's the relationship between the Riemannian metric and Jacobi field?

By Gauss lemma,
$$
\newcommand{\rd}{\mathrm d}
\rd s^2=(\rd r)^2+\tilde h_{ij}(r)(\theta)\cdot \rd\theta^i\cdot\rd\theta^j,
$$
where $\tilde h_{ij}(r)$ is a Riemannian metric on the sphere that ...

2
votes

### Let $M$ be a manifold with a conformal structure and a volume measure. How can one reconstruct a metric on $M$ from this data in a geometric way?

A conformal structure with Lorentzian representative produces a conformally invariant causal structure. Causal structure + volume = unique metric is claimed in Bombelli and Meyer (page 2) 1976. I'm ...

1
vote

### Let $M$ be a manifold with a conformal structure and a volume measure. How can one reconstruct a metric on $M$ from this data in a geometric way?

A conformal frame for a conformal structure $\sigma$ of signature $p,q$ on a manifold $M$ of dimension $n=p+q$ is a pair $(m,u)$ of point $m\in M$ and linear isomorphism $T_m M\xrightarrow{u}\mathbb{R}...

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