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 ...
0
votes
0answers
46 views

Parallel transport of a vector along a closed curve in curvilinear coordinates

There is an expression indicating the change of the vector parallel translation along a closed infinitesimal curve in curvilinear coordinates (one way of introducing curvature tensor): $$ \Delta A_{k} ...
7
votes
1answer
99 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
91 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
117 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 ...