Coherent tangent bundles and GaussBonnet formulas for wave fronts
Abstract
We give a definition of `coherent tangent bundles', which is an intrinsic formulation of wave fronts. In our application of coherent tangent bundles for wave fronts, the first fundamental forms and the third fundamental forms are considered as induced metrics of certain homomorphisms between vector bundles. They satisfy the completely same conditions, and so can reverse roles with each other. For a given wave front of a 2manifold, there are two GaussBonnet formulas. By exchanging the roles of the fundamental forms, we get two new additional GaussBonnet formulas for the third fundamental form. Surprisingly, these are different from those for the first fundamental form, and using these four formulas, we get several new results on the topology and geometry of wave fronts.
 Publication:

arXiv eprints
 Pub Date:
 October 2009
 arXiv:
 arXiv:0910.3456
 Bibcode:
 2009arXiv0910.3456S
 Keywords:

 Mathematics  Differential Geometry;
 57R45;
 53A05
 EPrint:
 23 pages, 7 figures This version 3 consists of sect. 1 &