The curvature of a Hessian metric

Abstract

Following P. M. H. Wilson's paper on sectional curvatures of Kahler 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 by Wilson about when this metric has nonpositive curvature. Also, we exhibit a large class of polynomials f on R^3 such that the associated metric has constant negative curvature. Dubrovin found another such polynomial on R^3, the Maschke sextic. We ask if these examples are the only ones with constant negative curvature, and we prove some partial results. This question is related both to the WDVV equations from string theory and to the "Clebsch covariant" from 19th-century invariant theory.