Lifts of maps to frame bundles

TL;DR

This paper investigates how maps between Riemannian manifolds can be lifted to their frame bundles, analyzing properties like conformality and harmonicity of these lifts under Mok metrics.

Contribution

It introduces a method to lift maps to frame bundles for non-diffeomorphic maps and studies their conformality and harmonicity properties.

Findings

Lifts of submersions can be constructed using horizontal distributions.

Conformality of lifts depends on the original map and the induced metrics.

Harmonicity of lifts relates to the harmonicity of the original map.

Abstract

Let $(M,g)$ be a Riemannian manifold, $L(M)$ be its frame bundle, $O(M)$ its orthonormal frame bundle. For a distribution $D$ on $M$ we define a subbundle $L(D)\subset L(M)$ or $O(D)\subset O(M)$ in a natural way. This allows us to consider a lift $L\varphi$ of a map $\varphi:M\to N$ not necessarily being a local diffeomorphism. More precisely, if $\varphi:M\to N$ is a submersion, then $L\varphi:L(\mathcal{H}^{\varphi})\to L(N)$ or $L\varphi:O(\mathcal{H}^{\varphi})\to L(N)$, where $\mathcal{H}^{\varphi}$ is a horizontal distribution of $\varphi$. Equipping $L(M)$ and $L(N)$ with the Mok metrics, we study conformality and harmonicity of lifts $L\varphi$.