# Algebraic conditions for the positivity of sectional curvature

## TL;DR

This paper investigates algebraic criteria ensuring the positivity of sectional curvature in Riemannian geometry, providing specific conditions for four-dimensional cases and insights into higher dimensions.

## Contribution

It offers a complete algebraic characterization of sectional positivity for Riemann curvature operators in four dimensions and discusses conditions in higher dimensions.

## Key findings

- Complete characterization for 4D curvature operators.
- Sufficient algebraic conditions for sectional positivity in 4D.
- Initial exploration of higher-dimensional curvature operators.

## Abstract

We examine algebraic conditions for the sectional positivity of the Riemann curvature operator. We describe sufficient conditions for dimension $n=4$, and complete characterization for a dense open subset of the space of operators in dimension $4$. We also briefly examine higher-dimentional curvature operators.