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.
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Communicated by Michael Taylor.
Dedicated to Professor Toshiki Mabuchi on the occasion of his sixtieth birthday.
K. Saji, M. Umehara and K. Yamada were partially supported by Grant-in-Aid for Scientific Research (Young Scientists (B)) No. 20740028, (A) No. 22244006 and (B) No. 21340016, respectively from the Japan Society for the Promotion of Science.
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Saji, K., Umehara, M. & Yamada, K. Coherent Tangent Bundles and Gauss–Bonnet Formulas for Wave Fronts. J Geom Anal 22, 383–409 (2012). https://doi.org/10.1007/s12220-010-9193-5
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DOI: https://doi.org/10.1007/s12220-010-9193-5