Golden and Metallic Structures on Hessian Manifolds

Abstract

We consider the reciprocal cost function $J(x)=\frac{1}{2}(x+x^{-1})-1$ and its $n$-dimensional extension $J(x_1,\dots ,x_n)=\frac{1}{2}(R+R^{-1})-1,R=\prod _{i=1}^n{x}_i^{{\alpha }_i},\alpha =({\alpha }_1,\dots ,{\alpha }_n)\in {\mathbb{R}}^n\setminus \{0\}.$ In logarithmic coordinates $t_i=\log x_i$, the Hessian of $J$ has a rank of one at every point. The associated Hessian geometry is degenerate and does not define a Riemannian metric. To obtain a nondegenerate geometric structure, we introduce a family of Hessian metrics $h_{\lambda }$. Combining the rank-one tensor with the Hessian metric $h_{\lambda }$, we construct a $(1,1)$-tensor field $A_{\lambda }$. Its trace normalization defines a projector $P_{\lambda }$, which induces an almost product structure and the corresponding golden and metallic structures. We study several properties of the projector $P_{\lambda }$ and the induced structures, including eigendistributions, parallelism, integrability, and curvature. The construction is given in an arbitrary dimension, and explicit formulas are obtained in the two-dimensional case. In particular, we show that the projector $P_{\lambda }$ is generally not parallel with respect to either the canonical flat affine connection or the Levi-Civita connection ${\nabla }^{\lambda }$ of the Hessian metric $h_{\lambda }$.