**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..