On the areas of geodesic triangles on a surface

Abstract

This paper treats geodesic triangles on two-dimensional orientable Riemannian manifolds $M$. Fixing two vertices $A$ and $B$, we can consider the area and the interior angles of the geodesic triangle $\Delta PAB$ as smooth functions of $P$. Applying the Laplace operator to these functions, we obtain formulas for the area and interior angles of $\Delta APAB$. It is shown that if $M$ is of constant curvature, the area and interior angles of geodesic triangles are harmonic.