On geodesic triangles with right angles in a dually flat space

TL;DR

This paper explores the properties of geodesic triangles with right angles in dually flat spaces, revealing how their interior angles can vary and establishing methods to construct such triangles with specific right-angle configurations.

Contribution

It introduces the study of geodesic triangles in Bregman manifolds, demonstrating constructions with right angles and dual Pythagorean theorems in non-self dual spaces.

Findings

Geodesic triangles with right angles can be constructed in dually flat spaces.

Dual Pythagorean theorems can hold simultaneously at a point for specific triples.

Interior angles of geodesic triangles can sum to π or exhibit excesses/defects.

Abstract

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 $\pi$ 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.