On geodesic triangles with right angles in a dually flat space
This work addresses a theoretical problem in information geometry for researchers studying geometric structures in statistical manifolds, and it appears incremental as it extends known concepts to non-self-dual cases.
The paper tackles the problem of constructing geodesic triangles with right angles in dually flat spaces (Bregman manifolds), showing how to build triangles with one, two, or three right angles and constructing point triples where dual Pythagorean theorems hold simultaneously.
The dualistic structure of statistical manifolds in information geometry yields eight types of geodesic triangles passing through three given points, the triangle vertices. The interior angles of geodesic triangles can sum up to $π$ like in Euclidean/Mahalanobis flat geometry, or exhibit otherwise angle excesses or angle defects. In this paper, we initiate the study of geodesic triangles in dually flat spaces, termed Bregman manifolds, where a generalized Pythagorean theorem holds. We consider non-self dual Bregman manifolds since Mahalanobis self-dual manifolds amount to Euclidean geometry. First, we show how to construct geodesic triangles with either one, two, or three interior right angles, whenever it is possible. Second, we report a construction of triples of points for which the dual Pythagorean theorems hold simultaneously at a point, yielding two dual pairs of dual-type geodesics with right angles at that point.