1
vote
1answer
57 views

Definition of CAT(0) metric space

I have a question regarding the definition of CAT(0) spaces. I am using the following definition: $X$ complete metric space is CAT(0) if $\forall z,y \in X$, $\exists m \in X$ such that $\forall x ...
1
vote
1answer
34 views

Metric spaces and curvature

Suppose that it is given that the Riemann curvature tensor in a metric space of dimension $d\geq2$ can be written as $$R_{abcd}=k(x^a)(g_{ac}g_{bd}-g_{ad}g_{bc})$$ where $x^a$ is a vector in the ...
8
votes
1answer
124 views

Hyperbolic diameter of Amsler's surface

I've recently learned about Amsler's surface, a surface of constant negative Gaussian curvature. If I understand things correctly, there is a whole family of such surfaces, differing in the angle of ...
2
votes
0answers
103 views

What is the correct distance measure for the (anti) de-Sitter space?

Given these two expressions 1) $\sinh{d}=\frac{\sqrt{t^2−x^2}}{\sqrt{1−(t^2−x^2)}}$ 2) $\sin{d}=\frac{\sqrt{t^2−x^2}}{\sqrt{1+(t^2−x^2)}}$ for distance $d$ from the origin $(0,0)$ to point $(x,t)$, ...
2
votes
0answers
129 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 ...