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Convex algebraic geometry of curvature operators. (English) Zbl 1475.14112

Summary: 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 \(n\geq5\), 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 \(n\geq5\), algorithms using semidefinite programming are obtained from hierarchies of inner approximations by spectrahedral shadows and outer relaxations by spectrahedra.

MSC:

14P10 Semialgebraic sets and related spaces
53B20 Local Riemannian geometry
53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions
90C22 Semidefinite programming
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