Title: Relative Chern character number and super-connection,
Abstract: For two complex vector bundles admitting a homomorphism, whose singularity
locates in the disjoint union of some odd--dimensional spheres, we give a
formula to compute the relative Chern characteristic number of these two
complex vector bundles. In particular, for a spin manifold admitting some
sphere bundle structure, we give a formula to express the index of a special
twisted Dirac operator.