New answers tagged riemannian-geometry
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 $W$ ...
- 41.6k
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,...
- 1,423
0
votes
Existence of adjoint operators on manifolds
You prove this as follows: As a first step you show the existence of the formally adjoint differential operator on a small open set $U$. This set $U$ is chosen, such that it carries a chart and such ...
- 754
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 ...
- 41.6k
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 ...
- 8,804
0
votes
Curvature of curves through a point of a surface smoothly embedded in Euclidean space
Let $u$ be a unit tangent vector at $p$.
The curvature $k$ in the direction of $u$ is $|\mathrm{I\!I}_p(u,u)|$, where $\mathrm{I\!I}_p$ is the second fundamental form at $p$;
it is a quadratic form on ...
- 41.6k
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 ...
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 ...
- 28.3k
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 ...
- 1,193
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}...
- 24.6k
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 ...
- 470
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