Coherent Tangent Bundles and Gauss–Bonnet Formulas for Wave Fronts

@article{Saji2009CoherentTB,
  title={Coherent Tangent Bundles and Gauss–Bonnet Formulas for Wave Fronts},
  author={Kentaro Saji and Masaaki Umehara and Kotaro Yamada},
  journal={Journal of Geometric Analysis},
  year={2009},
  volume={22},
  pages={383-409}
}
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… 
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