Geometric structures on Weil bundles: Canonical differential-geometric constructions

TL;DR

This paper explores how classical geometric structures on smooth manifolds can be canonically transferred to Weil bundles, providing a unified framework for rich geometric constructions and highlighting new non-trivial examples.

Contribution

It introduces a systematic method for lifting various geometric structures to Weil bundles, expanding the understanding of their differential geometric properties and applications.

Findings

Canonical lifts of structures like symplectic, contact, and Riemannian to Weil bundles.

Construction of a non-trivial cosymplectic Weil bundle example.

Preservation of integrability and characteristic vector fields under lifting.

Abstract

This paper investigates the transfer of classical geometric structures from a smooth manifold $M$ to its Weil bundle $(M^\mathbf A, \tilde\pi_M, M)$ associated with a Weil algebra $\mathbf A$. We show that various structures including locally conformal symplectic (lcs), locally conformal cosymplectic (lcc), contact, Jacobi, Sasakian, Walker, sub Riemannian, orientation, Riemannian, and K\"ahlerian structures admit canonical lifts to $M^\mathbf A$. Our approach emphasizes the differential geometric properties of these canonical constructions, utilizing the Weil projection $\tilde{\pi}_M$ and related functorial tools. This provides a unified perspective on endowing Weil bundles with rich geometric structure inherited from the base manifold. Furthermore, we highlight a specific construction yielding a cosymplectic manifold on $M^\mathbf{A}$ (for suitable $M$ and $\mathbf{A}$) that is demonstrably not a trivial suspension of a symplectic manifold. We also explicitly show how integrability of almost complex structures is preserved and clarify the nature of lifted characteristic vector fields.