Intrinsic Holmes-Thompson volumes and rigidity in Weil bundles

This paper develops a framework for defining intrinsic volumes on manifolds M by leveraging the structure of Weil bundles MA associated with Weil algebras A. We explore constructions for a Finsler-like structure FA primarily on the fibers of MA, aiming to derive it from the algebraic properties of A with minimal reliance on auxiliary metrics on M. The concept of A-naturality is introduced to formalize the intrinsic nature of such structures. From this fiberwise FA, an effective Finsler structure FM on the tangent bundle TM is derived. The Busemann-Hausdorff measure dVF associated with FM then provides a volume form on M. We establish foundational results concerning conditions under which a diffeomorphism ø: M → M preserves dVF, linking this to the behavior of its prolongation øA and exploring resulting rigidity phenomena, including a characterization theorem for dVF under affine symmetries. Furthermore, we propose several significant conjectures and future research directions concerning infinitesimal symmetries, axiomatic uniqueness of these volumes, interactions with curvature, sub-Riemannian limits, and holonomy restrictions.

کلیدواژه ها:

Weil Bundles ، Finsler Geometry ، Holmes-Thompson Volume ، Geometric Rigidity ، Affine Transformations