The curvature of a Hessian metric. (English) Zbl 1058.53032

Totaro, Burt

Int. J. Math. 15, No. 4, 369-391 (2004).

In the very clearly written paper the author considers a natural Riemannian metric on a hypersurface in a real vector space, defined by the Hessian of a homogeneous polynomial and gives a contribution to the question under which conditions this metric has nonpositive curvature. Among others he shows the following results:
(a) For the real quartic form \(f= xyz(x+ y+ z)\), the index cone in \(\mathbb{R}^3\) is nonempty, and the surface \(M\) inside the index cone has positive curvature everywhere.
(b) For the real cubic form \(f= (x^2{}_0+ x^2{}_1- x^2{}_2- x^2{}_3) x_3\), the index cone in \(\mathbb{R}^4\) is nonempty, and the 3-manifold \(M\) inside the index cone has positive sectional curvature on some 2-plane at every point.
The problem of finding all forms \(f\) on \(\mathbb{R}^3\) such that the surface \(M\) has constant curvature also has a close relation to classical invariant theory, in particular to the Clebsch covariant. The author proves that any ternary form \(f\) of degree at most 4 such that the surface \(M\) has constant curvature is in the closure of the set of forms which can be written as \(f= \alpha(x,y)+ \beta(z)\) in some linear coordinate system and generalizes in this way a well-known result of P. M. H. Wilson. The paper is completed by comprehensive references.

Reviewer: Hans Sachs (Leoben)

MSC:

53C20	Global Riemannian geometry, including pinching
15A72	Vector and tensor algebra, theory of invariants
53C21	Methods of global Riemannian geometry, including PDE methods; curvature restrictions

Keywords:

Hessian metric; centroaffine metric; Kähler moduli; Clebsch covariant

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