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A statistical manifold $(M,g,\nabla)$ is a Riemannian manifold with a torsion-free affine connection $\nabla$ such that $\nabla g$ is symmetric in all entries. Equivalently, there is a dual affine connection $\nabla^*$ such that $X(g(Y,Z)) = g(\nabla_X Y,Z) + g(Y,\nabla^*_X Z)$, with $X,Y,Z \in \mathcal{X}(M)$. See e.g. [1] for details, other associated tensors and equivalences.

In [2], there is a definition of homogeneous statistical manifold (Definition 1.4.1). Under suitable assumptions, the $G$-action induces a quotient (homogeneous) statistical structure $(G/H, \bar g, \bar \nabla)$. Is there a characterization of the non-trivial statistical structures on $G$ such that the induced structure on $G/H$ is trivial (i.e. $\bar \nabla$ is Levi-Civita, or the tensor $\bar \nabla \bar g$ vanishes)?

I was thinking in the related case of a product $(M_1 \times M_2, g_1 \times g_2, \nabla)$, with $G$ acting only on $M_2$, and where $(M_1,g_1,\nabla^1)$ is trivial, $(M_2, g_2, \nabla^2) $ is non-trivial, and $\nabla_{X_1+X_2} (Y_1+Y_2) := \nabla^1_{X_1} (Y_1) + \nabla^2_{X_2} (Y_2)$, with $X_i,Y_i \in \mathcal{X}(M_i)$. A priori there are alternative formulas for $\nabla$, right? I remember that in do Carmo's book (Exercise 1, Chapter 6), it is asked to show that the $\nabla$ above is the Levi-Civita connection on the product (with product metric) if the factors are Levi-Civita.

References

[1] Opozda B., Bochner’s technique for statistical structures, Annals of Global Analysis and Geometry, volume 48, 357–395 (2015) (https://arxiv.org/abs/1504.06307)

[2] Furuhata, Inoguchi, and Kobayashi - A characterization of the alpha-connections on the statistical manifold of normal distributions (https://arxiv.org/abs/2005.13927)

EDIT: change in the title and other minor changes.

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