The second variational formula of the variational problems for mappings between statistical manifolds

TL;DR

This paper derives the second variational formula for harmonic mappings between statistical manifolds, introduces stability concepts, and analyzes their properties, including examples of index and nullity calculations.

Contribution

It presents the second variational formula for harmonic mappings between statistical manifolds and explores stability, index, and nullity, extending previous first variational results.

Findings

Derived the second variational formula for harmonic mappings

Established stability, index, and nullity concepts for these mappings

Showed weak stability in non-positive curvature cases

Abstract

Recently, H. Furuhata and R. Ueno obtained the first variational formula of smooth mappings of a compact statistical manifold into another manifold. In this paper, we show the second variational formula of harmonic mappings of a statistical manifold into another one. Furthermore, we define the stability, the index and the nullity for the mappings, and we show the weakly stability for the harmonic mappings into any statistical manifolds of non-positive curvature. We also give several examples calculating the indexes and nullities.