A generalization of Pythagoras on a surface

Abstract

We analyze Toponogov's sine theorem for an infinitesimal geodesic triangle ABC on a C^2 regular surface M, which is given in his book [6, Problem 3.7.2] and we provide a generalization of the law of cosines for ABC on M. By replacing in the law of cosines B=\fracπ{2} on M, we derive the generalized theorem of Pythagoras on a surface: AC^2 = AB^2 + BC^2 + f(\angle A,\fracπ{2},AB,BC)o(AC^2) or AC^2 = AB^2 + BC^2 + (\angle A + \angle C-\fracπ{2})^2 where f(\angle A,\angle B,AB,BC) is a rational function w.r. to cosA; cosB, sinA, sinB, AB and BC.

6 pages