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≥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≥5, algorithms using semidefinite programming are obtained from hierarchies of inner approximations by spectrahedral shadows and outer relaxations by spectrahedra.
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