The dualistic structure of statistical manifolds  yields eight types of geodesic triangles defining overall eighteen interior angles. In general, the interior angles of geodesic triangles can sum up to \pi like in Euclidean/Mahalanobis  geometry, or exhibit either angle excesses or angle defects. In this paper, we initiate the study of geodesic triangles in dually flat spaces  where a generalized Pythagorean theorem holds.  First, we show when it is possible how to construct geodesic triangles which either have one, two, or three interior right angles . Then 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 geodesic triangles with doubly right angles .
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 geodesics with right angles at that point .
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in information geometry
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defining overall eighteen interior angles. In general, the
passing through three given points, the triangle vertices. The
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flat
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either
otherwise
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, termed Bregman manifolds,
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We consider non-self dual Bregman manifolds since Mahalanobis self-dual manifolds amount to Euclidean geometry.
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when it is possible
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which either have
with either
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. Then
, whenever it is possible. Second,
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geodesic triangles with doubly right angles
geodesics with right angles at that point