Coherent tangent bundles and Gauss-Bonnet formulas for wave fronts

Details Coherent tangent bundles and Gauss-Bonnet formulas for wave fronts Coherent tangent bundles and Gauss-Bonnet formulas for wave fronts

2009-10-19

arxiv.org/abs/0910.3456v3

Mathematics

Differential Geometry

23 pages, 7 figures This version 3 consists of sect. 1 & 4 of the previous version whose title was "The intrinsic duality of w

Scientific paper

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 2-manifold, there are two Gauss-Bonnet formulas. By exchanging the roles of the fundamental forms, we get two new additional Gauss-Bonnet 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.

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