The Question
Given Hessian manifold $M$, there is a natural Kahler structures on $TM$. Is it possible to write the Kahler potential of these in terms of the Hessian potential?
Background
To elaborate a bit more, let me give some background information. There are two separate definitions for a Hessian manifold $(M,g)$ which initially appear distinct but turn out to be the same.
- There are local coordinates $\{x_i \}_{i=1}^n$ and a convex potential $\psi$ so that in the $x$-coordinates, $$g_{ij}= \frac{ \partial^2 \psi}{\partial x_i x_j} $$
- $(M,g)$ locally admits a dually flat structure. That is to say, it admits two flat connections $\nabla$ and $\nabla^*$ satisfying $$X(g(Y,Z)) = g(\nabla_X Y,Z) + g(Y, \nabla^*_X Z). $$
If we consider the tangent bundle of $TM$, we can use the flat connection $\nabla$ to induce an almost complex structure $J^\nabla$ and a Sasaki metric $g^\nabla$ on $TM$. It turns out that $(TM, g^\nabla, J^\nabla)$ is a Kahler manifold if and only if $(M,g)$ is Hessian. It's worth noting that we can dualize all of this and obtain a second Kahler structure on $TM$ using the dual connection, as well.
In this setting, I'm wondering if it's possible to use potential $\psi$ to write the Kahler potential $\Psi$ for the Sasaki metric and induced complex structure.
For more details on the Sasaki metric and almost complex structure on $TM$, the paper of Satoh has more information.
Satoh, Hiroyasu, Almost Hermitian structures on tangent bundles, Suh, Young Jin (ed.) et al., Proceedings of the 11th international workshop on differential geometry, Taegu, Korea, November 9--11, 2006. Taegu: Kyungpook National University. 105-118 (2007). ZBL1125.53022..