Geometric structures on Weil bundles: Canonical differential-geometric constructions
S. Tchuiaga, A. Ndiaye, C. Khoule, R. A. M. Mohameden
TL;DR
The paper develops a unified framework to transfer classical differential-geometric structures from a smooth manifold $M$ to its Weil bundle $M^{\mathbf{A}}$ via the Weil functor $T^{\mathbf{A}}$, showing that many structure types are canonically lifted while preserving essential properties such as integrability and symmetry. It emphasizes the distinction between canonical lifts $T^{\mathbf{A}}$ and naive pullbacks, introduces averaged lifts using sections to handle Jacobi-type structures, and provides concrete constructions for lcs, lcc, cosymplectic, contact, Jacobi, Sasakian, Walker, sub-Riemannian, orientation, Riemannian, and Kähler geometries, including explicit Reeb/Killing-field lifts. A nontrivial cosymplectic example demonstrates that $M^{\mathbf{A}}$ can realize genuinely new geometries not arising from simple products, and integrability of almost complex structures is preserved under canonical lifts. The work also notes that curvature-related or Einstein-type properties are typically not preserved, and it outlines future directions in Floer theory, generalized geometry lifts, and curvature-flow analyses on Weil bundles.
Abstract
This paper investigates the transfer of classical geometric structures from a smooth manifold $M$ to its Weil bundle $(M^\mathbf A, \tildeπ_M, M)$ 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 Kählerian structures admit canonical lifts to $M^\mathbf A$. Our approach emphasizes the differential geometric properties of these canonical constructions, utilizing the Weil projection $\tildeπ_M$ 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.