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Journal of Geometric Analysis

, Volume 22, Issue 2, pp 383–409 | Cite as

Coherent Tangent Bundles and Gauss–Bonnet Formulas for Wave Fronts

  • Kentaro Saji
  • Masaaki Umehara
  • Kotaro YamadaEmail author
Article

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 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.

Keywords

Wave front Curvature The Gauss–Bonnet Formula Fold 

Mathematics Subject Classification (2000)

57R45 53A05 

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Copyright information

© Mathematica Josephina, Inc. 2010

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of EducationGifu UniversityGifuJapan
  2. 2.Department of Mathematics, Graduate School of ScienceOsaka UniversityOsakaJapan
  3. 3.Department of MathematicsTokyo Institute of TechnologyTokyoJapan

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