Golden and Metallic Structures on Hessian Manifolds

We consider the reciprocal cost function J(x)=12(x+x−1)−1 and its n-dimensional extension J(x1,…,xn)=12(R+R−1)−1,R=∏i=1nxiαi,α=(α1,…,αn)∈Rn∖{0}. In logarithmic coordinates ti=logxi, 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 hλ. Combining the rank-one tensor with the Hessian metric hλ, we construct a (1,1)-tensor field Aλ. Its trace normalization defines a projector Pλ, which induces an almost product structure and the corresponding golden and metallic structures. We study several properties of the projector Pλ 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 Pλ is generally not parallel with respect to either the canonical flat affine connection or the Levi-Civita connection ∇λ of the Hessian metric hλ.