# Non-negative Curvature and Conullity of the Curvature Tensor

## TL;DR

This paper investigates the geometric implications of curvature tensors with specific conullity conditions, showing they impose strong topological and geometric restrictions, especially under non-negative curvature assumptions.

## Contribution

It characterizes manifolds with conullity two and three under non-negative curvature, revealing they are either Euclidean or locally product spaces.

## Key findings

- Manifolds with conullity 2 and non-negative curvature are diffeomorphic to Euclidean space or products.
- Finite volume manifolds with conullity 3 are locally products.
- Conditions restrict the topology and geometry of the manifolds significantly.

## Abstract

The conullity of a curvature tensor is the codimension of its kernel. We consider the cases of conullity two in any dimension and conullity three in dimension four. We show that these conditions are compatible with non-negative sectional curvature only if either the manifold is diffeomorphic to $\mathbb{R}^n$ or the universal cover is an isometric product with a Euclidean factor. Moreover, we show that finite volume manifolds with conullity 3 are locally products.