Sharp sectional curvature bounds and a new proof of the spectral theorem

Calle, Maxine; Dunn, Corey

doi:10.2140/involve.2020.13.445

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Sharp sectional curvature bounds and a new proof of the spectral theorem

Maxine Calle and Corey Dunn

Vol. 13 (2020), No. 3, 445–454

DOI: 10.2140/involve.2020.13.445

Abstract

We algebraically compute all possible sectional curvature values for canonical algebraic curvature tensors and use this result to give a method for constructing general sectional curvature bounds. We use a well-known method to geometrically realize these results to produce a hypersurface with prescribed sectional curvatures at a point. By extending our methods, we give a relatively short proof of the spectral theorem for self-adjoint operators on a finite-dimensional real vector space.

Keywords

sectional curvature, canonical algebraic curvature tensor, spectral theorem

Mathematical Subject Classification 2010

Primary: 15A69

Secondary: 15A63, 53C21

Milestones

Received: 16 October 2019

Revised: 17 March 2020

Accepted: 28 April 2020

Published: 14 July 2020

Communicated by Frank Morgan

Authors

Maxine Calle
Reed College Portland, OR United States

Corey Dunn
Mathematics Department California State University at San Bernardino San Bernardino, CA United States

Vol. 13, No. 3, 2020