On the areas of geodesic triangles on a surface (Q5937624)

Given a geodesic triangle with vertices \( A, B, P \) in a two-dimensional Riemannian manifold \( M \), the authors study the area \( S(P) \) and the angle \( \Omega(P) \) at \( P \) as real functions of \( P \) for fixed \( A, B \). Suitable assumptions have to be made in order to assure that the triangle is (at least locally) uniquely determined by \( P \) and is compact. They calculate the Laplacians of \( S \) and \( \Omega \) in terms of the geodesic coordinates w.r.t. \( A \) and \( B \), giving rise to `bi-angular coordinates'. Specifically, the triangle should be `small' or `somewhat small and in good condition', where e.g. `small' means that there is an open convex set \( D \) containing \( A, B, P \). As a corollary, if \( M \) is of constant curvature, it is shown that \( S \) and \( \Omega \) are harmonic functions, thus generalizing a result of \textit{Hasebe, Kiso, Kaneda} and \textit{Doke} [Japan J. Appl. Phys. 32, 2162-2166 (1993)] found by explicit calculations on the sphere \( M = S^{2} \).