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

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.


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
Full Text: DOI arXiv


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