Lifting diffeomorphisms to vector bundles

TL;DR

This work investigates when a smooth diffeomorphism $\varphi:M\to M$ can be lifted to a linear automorphism of a vector bundle $p:V\to M$, i.e., when $\varphi$ lies in the image of $h_V:\mathrm{AUT}\,V\to \mathrm{Diff}\,M$. It identifies obstructions given by characteristic classes (Stiefel-Whitney for real bundles and Chern for complex bundles) and shows sufficiency in the natural/jet-frame setting, with the lifting problem depending only on the homotopy class of $\varphi$. The reduction to metric bundles clarifies how a lift reduces to lifting diffeomorphisms to isometries of $(V,\langle\cdot,\cdot\rangle)$, and polar decomposition yields a canonical isometry factor. The complex case is analyzed via a $Gl(r,\mathbb{C})$-bundle $E$ and a compatibility condition with the complex structure, with cohomological vanishing results guaranteeing complex-linear lifts in several manifolds. The paper concludes with detailed examples on Grassmannians, projective spaces, spheres, and Hopf fibrations, illustrating when and how lifts exist and providing explicit constructions.

Abstract

Criteria for a diffeomorphism of a smooth manifold $M$ to be lifted to a linear automorphism of a given real vector bundle $p\colon V\rightarrow M$, are stated. Examples are included and the metric and complex vector-bundle cases are also considered.

Theorems & Definitions (28)

• Proposition 1
• Lemma 2
• Proposition 3
• Corollary 4
• Proposition 5
• Corollary 6
• Proposition 7
• Corollary 8
• Remark 9
• Remark 10