THE CURVATURE OF A HESSIAN METRIC

Abstract

Inspired by Wilson's paper on sectional curvatures of Kähler moduli, we consider a natural Riemannian metric on a hypersurface {f=1} in a real vector space, defined using the Hessian of a homogeneous polynomial f. We give examples to answer a question posed by Wilson about when this metric has nonpositive curvature. Also, we exhibit a large class of polynomials f on R3 such that the associated metric has constant negative curvature. We ask if our examples, together with one example by Dubrovin, are the only ones with constant negative curvature. This question can be rephrased as an appealing question in classical invariant theory, involving the "Clebsch covariant". We give a positive answer for polynomials of degree at most 4, as well as a partial result in any degree.