Convex Algebraic Geometry of Curvature Operators

Abstract

We study the structure of the set of algebraic curvature operators satisfying a sectional curvature bound under the light of the emerging field of Convex Algebraic Geometry. More precisely, we determine in which dimensions $n$ this convex semialgebraic set is a spectrahedron or a spectrahedral shadow; in particular, for $n\ge 5$, these give new counter-examples to the Helton--Nie Conjecture. Moreover, efficient algorithms are provided if $n=4$ to test membership in such a set. For $n\ge 5$, algorithms using semidefinite programming are obtained from hierarchies of inner approximations by spectrahedral shadows and outer relaxations by spectrahedra.

LaTeX2e, 28 pages, final (revised) version. To appear in SIAM J. Appl. Algebra Geom