# Intrinsic vs. extrinsic; also Pogorelov vs. me

There are two common ways to measure distances between two points on a sphere, such as the (idealized) surface of the Earth:

• intrinsic: length of the shortest path along the surface (not coincidentally, such a path is called a geodesic)
• extrinsic: length of the chord between two points (never mind that traveling along this chord requires digging a tunnel).

The same choice between intrinsic and extrinsic metrics is present for any surface. It is not hard to recover the intrinsic distance from the extrinsic one: $(*)\qquad\qquad\displaystyle d_i(p,q)=\lim_{\delta\to 0}\inf \left\{ \sum_{k=1}^n d_e(x_k,x_{k-1})\colon x_0=p, x_n=q, \; \forall k\; d_e(x_k,x_{k-1})<\delta \right\}$

The construction (*) mimics the Riemannian definition of length in which the integrals along curves from $p$ to $q$ are replaced with sums along chains of points from $p$ to $q$. In the process, the surface forgets about its particular embedding into Euclidean space. For example, the parabolic cylinder $z=x^2$ and the plane $z=0$ have different extrinsic metrics, but their intrinsic metrics are the same (in the sense that there is a bijection between the two surfaces which preserves the intrinsic metric).

However, I would be hard pressed to find such an example with the paraboloid $z=x^2+y^2$ or a sphere. In fact, Pogorelov’s uniqueness theorem states that the intrinsic metric determines extrinsic metric for any closed convex surface, as well as for unbounded closed surfaces of total curvature at least $2\pi$. In all likelihood, the problem was posed to Pogorelov by his advisor A.D. Alexandrov, and Pogorelov’s book Extrinsic Geometry of Convex Surfaces is obviously meant as a companion to Alexandrov’s Intrinsic Geometry of Convex Surfaces.

Pogorelov is not exactly a household name in the Western hemisphere, even though his work on geometry of surfaces and on the Monge–Ampère equation eventually came to be cited. (The 4th Hilbert problem story is complicated, and perhaps will make another blog post.) The situation is remarkably different in Russia where Pogorelov’s “Geometry 6-10” was the standard geometry textbook for a long time. The numbers 6-10 refer to the grades from 6th to 10th. With the switch to 11-grade system these became “7-11”.

Proof. Let $ABC$ be a triangle in which $\angle A=\angle B$ (Fig.50). We will prove that it is isosceles with base $AB$.
The triangle $ABC$ is equal to the triangle $BAC$ by the second criterion of equality of triangles. Indeed, $AB=BA$, $\angle B=\angle A$, $\angle A=\angle B$. The equality of triangles implies that $AC=BC$. Therefore, $ABC$ satisfies the definition of isosceles triangle. Theorem is proved.