We determine the index and co-index of the twisted tangent bundle of projective spaces. We also discuss the stabilty of them, and determine the set of integers that can be realized as the stable co-index of a vector bundle over the projective space.
In this paper we shall prove the existence of sharp remainder terms involving singular weight (logR/|x|)-2 for Hardy-Sobolev inequalities of the following type: ∫Ω|∇u(x)|2dx≥(n-2/2)2∫Ω|u(x)|2/|(x)|2dx for any u∈W1, 20(Ω), Ω is a bounded domain in Rn, n>2, with 0∈Ω. Here the number of remainder terms depends on the choice of R.
The object of our research is a piecewise Riemannian 2-polyhedron which is a combinatorial 2-polyhedron such that each 2-simplex is isometric to a triangle bounded by three smooth curves on some Riemannian 2-manifold. In the previous paper , which is a joint work with J. Itoh, we have introduced the concept of total curvature for piecewise Riemannian 2-polyhedra and proved a generalized Gauss-Bonnet theorem and a generalized Cohn-Vossen theorem. In this paper, we shall give a definition of flatness of piecewise Riemannian 2-polyhedra and characterize them.