Geometry in the large on Hadamard manifolds

TL;DR

This paper establishes new Liouville-type theorems demonstrating the non-existence of certain geometric tensors on Hadamard manifolds, extending classical vanishing results to non-compact symmetric spaces.

Contribution

It introduces novel Liouville-type theorems for tensors on Hadamard manifolds, expanding the scope of vanishing theorems beyond compact cases.

Findings

Proved non-existence of Killing-Yano tensors on Hadamard manifolds

Established vanishing results for harmonic symmetric tensors

Extended Liouville theorems to non-compact symmetric spaces

Abstract

In this paper, we prove several Liouville-type theorems on the non-existence of Killing-Yano tensors, Killing tensors, and harmonic symmetric tensors on Hadamard manifolds and, in particular, on Riemannian symmetric spaces of non-compact type. These theorems supplement the well-known vanishing theorems for the above tensors, obtained using the Bochner technique for compact Riemannian manifolds. In turn, the proofs of our theorems will use well-known Liouville-type theorems on the non-existence of subharmonic and harmonic functions on complete Riemannian manifolds, which we have partially modified for the case of Hadamard manifolds and, in particular, Riemannian symmetric spaces of noncompact type.