Sharp sectional curvature bounds and a new proof of The Spectral Theorem

TL;DR

This paper algebraically determines sectional curvature bounds for canonical tensors, constructs hypersurfaces with prescribed curvatures, and provides a new, concise proof of the Spectral Theorem for self-adjoint operators.

Contribution

It introduces a novel algebraic approach to sectional curvature bounds and offers a new proof of the Spectral Theorem, connecting curvature analysis with spectral theory.

Findings

Computed all sectional curvature values for canonical algebraic curvature tensors

Developed a method to construct hypersurfaces with specific sectional curvatures

Provided a shorter proof of the Spectral Theorem for self-adjoint operators

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.