Tubular neighborhoods play an important role in modern differential topology. The main aim of the paper is to apply these constructions to geometry of structures on Riemannian manifolds. Deformations of tensor structures on a normal tubular neighborhood of a submanifold in a Riemannian manifold are considered in section 1. In section 2, this approach is used to obtain a Kählerian structure on the corresponding normal tubular neighborhood of the null section in the tangent bundle TM of a smooth manifold M. In section 3, we consider a new deformation of a tensor structure on some neighborhood of a curve and introduce the so-called geometric antigravitation. Some results of the paper were announced in , . The work  is close to our discussion.