On a constant curvature statistical manifold. (English) Zbl 1497.53038
Summary: We will show that a statistical manifold \((M, g, \nabla)\) has a constant curvature if and only if it is a projectively flat conjugate symmetric manifold, that is, the affine connection \(\nabla\) is projectively flat and the curvatures satisfies \(R=R^*\), where \(R^*\) is the curvature of the dual connection \(\nabla^*\). Moreover, we will show that properly convex structures on a projectively flat compact manifold induces constant curvature \(-1\) statistical structures and vice versa.
MSC:
| 53B12 | Differential geometric aspects of statistical manifolds and information geometry |
| 53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |
Keywords:
statistical manifolds; constant curvatures; conjugate symmetries; projective flatness; properly convex structuresReferences:
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