Two-root Riemannian Manifolds

TL;DR

This paper introduces the concept of k-root manifolds, focusing on one-root and two-root cases, and characterizes two-root Riemannian manifolds, including their existence and algebraic curvature tensors.

Contribution

It defines k-root manifolds based on eigenvalues of the Jacobi operator and classifies two-root Riemannian manifolds, extending the theory of locally two-point homogeneous spaces.

Findings

No two-root Riemannian manifold exists in odd dimensions.

All two-root Riemannian algebraic curvature tensors are described in twice an odd dimension.

Additional conditions for two-root Riemannian manifolds are provided.

Abstract

Osserman manifolds are a generalization of locally two-point homogeneous spaces. We introduce $k$-root manifolds in which the reduced Jacobi operator has exactly $k$ eigenvalues. We investigate one-root and two-root manifolds as another generalization of locally two-point homogeneous spaces. We prove that there is no two-root Riemannian manifold of odd dimension. In twice an odd dimension, we describe all two-root Riemannian algebraic curvature tensors and give additional conditions for two-root Riemannian manifolds.