Which singular tangent bundles are isomorphic?

Abstract

Logarithmic and $b$-tangent bundles provide a versatile framework for addressing singularities in geometry. Introduced by Deligne and Melrose, these modified bundles resolve singularities by reframing singular vector fields as well-behaved sections of these singular bundles. This approach has gained significant attention in symplectic geometry, particularly through its applications to the study of Poisson manifolds that are symplectic away from a hypersurface ($b^m$-symplectic forms). In this article, we investigate the conditions under which these singular tangent bundles are isomorphic to the tangent bundle or other singular bundles, analyzing in detail the low-dimensional case and the case of spheres. We also examine the existence of geometric structures in light of these conditions. Furthermore, we establish a Poincaré-Hopf theorem for the $b^m$-tangent bundle, offering new insights into the interplay between singular structures and topological invariants.