Golden and Metallic Structures on Hessian Manifolds

Abstract

We consider the reciprocal cost function $ J(x)=\frac12(x+x^{-1})-1 $ and its $n$-dimensional extension $J(x_1,\ldots,x_n) = \frac12(R+R^{-1})-1, R=\prod\limits_{i=1}^n x_i^{\alpha_i}, \alpha=(\alpha_1,\ldots,\alpha_n)\in\mathbb{R}^n\setminus\{0\}.$ 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 $h_\lambda$.