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

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.