5
votes
2answers
137 views

Two distinct geodesics joining two points on a compact manifold

This is a problem from the book Gallot, Hulin, Lafontaine: Riemannian geometry (3rd edition). Exercise 2.118: For a compact Riemannian manifold, let $p,q$ two points such that $d(p,q) = ...
3
votes
1answer
60 views

Is a Riemannian metric a $2$-form?

In Lee's Riemannian Manifolds; An introduction to Curvature, he defines a Riemannian metric as an element of $\Gamma(T^2_0M)$, a $(2,0)$-tensor. Is this the same thing as a $2$-form? Is there a ...
3
votes
2answers
79 views

Riemannian manifolds are metrizable?

I've seen lots of casual claims that Riemannian manifolds (even without assuming second-countability) are metrizable. In the path-connected case, we can use arc-length to create a distance function. ...
0
votes
0answers
51 views

Interior of a Dirichlet domain in a Riemannian manifold

Let $X\neq\varnothing$ be a complete connected Riemannian manifold. Suppose $G$ is a group of isometries of $X$, acting properly discontinuously on $X$. We assume there is a point $x_0\in X$ such that ...
0
votes
1answer
54 views

How to solve for the extrinsic variables of a one variable scaled conformal metric to an equivalent metric?

Given the following metric equivalence \begin{align} e^{2w(x_2)} \left( dx_1^2+dx_2^2 \right) = dy_1^2+dy_2^2+dy_3^2 \end{align} is their a known solution for the extrinsic variables $y_1(x_1,x_2)$, ...
2
votes
1answer
22 views

Local geodesics in uniquely geodesic spaces

Suppose $Y$ is a proper, uniquely geodesic metric space. In such a space, is any local geodesic in fact a geodesic? Here the terms "geodesic" and "local geodesic" are taken in the metric sense: a ...
5
votes
1answer
87 views

Is a bounded (Riemannian) manifold always totally bounded?

Let $(M,g)$ be a finite dimensional Riemannian manifold. Suppose it is bounded as a metric space, i.e. $$\sup \{ d(x,y) | x,y \in M \} < \infty$$ where $d$ is the distance induced by $g$. Is $M$ ...
0
votes
1answer
60 views

Metric completion of universal covering of punctured plane

It is known that the universal covering of the punctured plane $\mathbb C\setminus\{0\}$ is $\exp:\mathbb C\to\mathbb C\setminus\{0\}$. In real coordinates, $f=\exp:\tilde M=\mathbb R^2\to M=\mathbb ...
0
votes
1answer
55 views

embedding discrete metric into manifold?

True or false: "Any edge-weighted undirected graph can be isometrically embedded into some Riemannian manifold". "isometric embedding" here means that for any pair of nodes, their shortest path ...
7
votes
1answer
135 views

Covering a Riemannian manifold with geodesic balls without too much overlap

I'm looking for a proof of the following fact: Let $M$ be a compact Riemannian manifold. There is a natural number $h$, such that for any sufficiently small number $r>0$, there exists a cover of ...
5
votes
1answer
114 views

Metric on Riemannian manifolds

Why is it necessary to consider taking the infimum over the lengths of all piece-wise smooth curves while defining the distance function on a Riemannian Manifold instead of just taking the infimum ...
1
vote
1answer
47 views

Reflection along subspace

A symmetric space is a Riemannian manifold M with the following property: For every point $p \in M$ there is an isometry $\phi: M \rightarrow M$ such that $\phi(p) = p$ and $\phi_*(v_p) = -v_p \in ...
1
vote
0answers
23 views

Inequality in geodesic quadrupel

Let $(M,g)$ be a compact complete Riemannian manifold. Consider the geodesic quadrupel $ABCD$ where $l:=d(A,B)=d(B,C)=d(C,D)=d(A,D) \geq \frac{1}{2}$ and $d(B,D),d(A,C) \geq 1$. Let $x \in CD$ and $y ...
4
votes
1answer
327 views

Ricci tensor for a 3-sphere without Math packets

Let's have the metric for a 3-sphere: $$ dl^{2} = R^{2}\left(d\psi ^{2} + sin^{2}(\psi )(d \theta ^{2} + sin^{2}(\theta ) d \varphi^{2})\right). $$ I tried to calculate Riemann or Ricci tensor's ...
3
votes
2answers
111 views

Minimal length of non-contractible loops

Not self-intersecting loops on a connected closed orientable smooth surface $S$ must have a minimal length not to disconnect it, e.g. the equators of a torus. "Not to disconnect" is - on such surfaces ...
4
votes
1answer
169 views

Relating volume elements and metrics. Does a volume element + uniform structure induce a metric?

AFAIK a metric uniquely determines the volume element up to to sign since the volume element since a metric will determine the length of supplied vectors and angle between them, but I do not see a way ...
2
votes
1answer
66 views

$\operatorname{Isom}{(M)}$ has Lie-structure for M metrizable manifold

Suppose $M$ is a smooth and metrizable manifold. Then $\operatorname{Isom}{(M)}$ can be given the structure of a Lie group, so that the action of $\operatorname{Isom}{(M)}$ on $M$ is still smooth. I ...
1
vote
1answer
126 views

A new metric involving curves

Let $(X, d)$ be a metric space. The inner metric or length metric associated with $d$ is the function $d_i : X \times X \to [0,\infty]$ defined by $$d_i(x, y) := \inf L(\sigma)$$ where the infimum is ...
2
votes
0answers
127 views

What's the relationship between the riemannian metric and Jacobi field?

I encounter to the question in reading the following Excise: Let $(M,g)$ be a $m$-dimensional Riemannian manifold, and $(r,\theta^1,\theta^2,\ldots,\theta^{m-1})$ be the (geodesic) polar ...
2
votes
0answers
237 views

the hyperbolic plane is complete

I'd like to prove that the hyperbolic plane is complete in a short, nice way. Here's my proof, but I'm not convinced of its correctness yet. Let $\mathbb{H}=\left\{(x,y)\in\mathbb{R}^2\big\vert ...
9
votes
3answers
344 views

Existence of a Riemannian metric inducing a given distance.

Let $M$ be a smooth, finite-dimensional manifold. Suppose $M$ is also a metric space, with a given distance function $d: M \times M \rightarrow \mathbb{R}_{+}$, which is compatible with the original ...
2
votes
2answers
261 views

metric on the universal covering of a geometric manifold

We know that the universal covering of a closed hyperbolic 3-manifold can be identified with the hyperbolic space $\mathbb{H}^3$. Now, what is not very clear to me is how this identification has to be ...