Cosine theorem in a Riemannian manifold

Question

In an euclidean space, given the rectangle with vertices $A,B,C$ the cosine theorem states: $\overline{AB}^2=\overline{BC}^2+\overline{AC}^2-2\overline{AC}\dot{}\overline{BC}\dot{}\cos(\widehat{AC BC})$. If we have a metric tensor $g_{\mu\nu}$ and so: $dx^2=g_{\mu\nu}dx^{\mu}dx^{nu}$ what is the new expression of the cosine theorem in the new space? For example, in the simple case of spherical trigonometry how does the theorem change? Thanks.

Answer 1

In the spherical case, you can look at the Wikipedia article, and similarly in the hyperbolic case. In general, you get something disgusting, so people have proved comparison theorems instead, which say that for a small triangle with sides $a, b, c$ in space $Y$ the angles are bigger than those in a "model space" $X,$ if the (sectional) curvature of $Y$ is bigger than that of $X.$ For more, search for Toponogov comparison theorem. (there is a huge area of research known as the study of CAT(?) spaces (? can be 0 or 1 or -1), which are spaces with curvature bounds. Just do a search for, e.g., CAT(0).