Convex Algebraic Geometry of Curvature Operators

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 counterexamples 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.