Why is it said that every point in hyperbolic space is a saddle point?

Question

I have read that since hyperbolic space has a constant negative curvature (a concept that I think I understand), every point is a saddle point. I am trying to understand what that means. Can we say anything similar about every point in elliptic space, since there we have constant positive curvature? What about points in Euclidean space?

Answer 1

Note that "curvature" in your sense refers to Gaussian curvature, an intrinsic measure of angular defect in geodesic triangles, while "saddle point" is an extrinsic concept, related to a particular embedding in $\mathbf{R}^3$.

These conditions are equivalent in the sense that the Gaussian curvature at a point $p$ of a smooth surface $S$ in $\mathbf{R}^3$ is equal to the product of the principle curvatures of $S$ at $p$, and this product is negative if and only if the principle curvatures at $p$ have opposite sign, if and only if $S$ is saddle-shaped at $p$ in the sense Will Jagy mentions.

Similarly, the Gaussian curvature of $S$ at $p$ is positive if and only if the principle curvatures of $S$ at $p$ have the same sign, if and only if $S$ is "bowl-shaped" at $p$.

If the Gaussian curvature of $S$ at $p$ is zero, then at least one principle curvature vanishes at $p$, and qualitatively you can't say much. For example, the graph of $f(x, y) = x^4 + ky^2$ has Gaussian curvature $0$ at the origin independently of $k$ (because the graph is quadratically flat along the $x$-axis), but the surface is (loosely) either bowl- or saddle-shaped, depending on the sign of $k$.

Answer 2

Being a saddle points means that the product of the principal curvatures is negative and vice versa.