# 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}
}
• Published 19 October 2009
• Mathematics
• Journal of Geometric Analysis
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…
27 Citations
The Gauss–Bonnet Theorem for Coherent Tangent Bundles over Surfaces with Boundary and Its Applications
• Mathematics
The Journal of Geometric Analysis
• 2019
In Saji et al. (J Math 62:259–280, 2008; Ann Math 169:491–529, 2009; J Geom Anal 222):383–409, 2012) the Gauss–Bonnet formulas for coherent tangent bundles over compact-oriented surfaces (without
An index formula for a bundle homomorphism of the tangent bundle into a vector bundle of the same rank, and its applications
• Mathematics
• 2012
In a previous work, the authors introduced the notion of coherent tangent bundle', which is useful for giving a treatment of singularities of smooth maps without ambient spaces. Two different types
Intrinsic properties of surfaces with singularities
• Mathematics
• 2014
In this paper, we give two classes of positive semi-definite metrics on 2-manifolds. The one is called a class of Kossowski metrics and the other is called a class of Whitney metrics: The pull-back
The duality of conformally flat manifolds
• Mathematics
• 2010
AbstractIn a joint work with Saji, the second and the third authors gave an intrinsic formulation of wave fronts and proved a realization theorem for wave fronts in space forms. As an application, we
Isometric Immersions with Singularities Between Space Forms of the Same Positive Curvature
In this paper, we give a definition of coherent tangent bundles of space form type, which is a generalized notion of space forms. Then, we classify their realizations in the sphere as a wave front,
Behavior of Gaussian curvature near non-degenerate singular points on wave fronts
• Mathematics
• 2013
We define cuspidal curvature $\kappa_c$ along cuspidal edges in Riemannian $3$-manifolds, and show that it gives a coefficient of the divergent term of the mean curvature function. Moreover, we show
A note on singular points of bundle homomorphisms from a tangent distribution into a vector bundle of the same rank
• Mathematics
Rocky Mountain Journal of Mathematics
• 2019
We consider bundle homomorphisms between tangent distributions and vector bundles of the same rank. We study the conditions for fundamental singularities when the bundle homomorphism is induced from
Complete flat fronts as hypersurfaces in Euclidean space
By Hartman--Nirenberg's theorem, any complete flat hypersurface in Euclidean space must be a cylinder over a plane curve. However, if we admit some singularities, there are many non-trivial examples.
Extrinsic diameter of immersed flat tori in S3
• Mathematics
• 2011
Enomoto, Weiner and the first author showed the rigidity of the Clifford torus amongst the class of embedded flat tori in S3. In the proof of that result, an estimate of extrinsic diameter of flat

## References

SHOWING 1-10 OF 20 REFERENCES
The Boy–Gauss–Bonnet Theorems for C∞-Singular Surfaces with Limiting Tangent Bundle
Using C∞-singularity theory, we describe natural conditions on nonimmersivemappings of a compact orientable 2-manifold into Euclidean 3-space (e.g., theexistence of a global limiting tangent bundle).
Singularities of flat fronts in hyperbolic space
• Mathematics
• 2004
It is well-known that the unit cotangent bundle of any Riemannian manifold has a canonical contact structure. A surface in a Riemannian 3-manifold is called a front if it is the projection of a
The duality between singular points and inflection points on wave fronts
• Mathematics
• 2009
In the previous paper, the authors gave criteria for AkC1-type singularities on wave fronts. Using them, we show in this paper that there is a duality between singular points and inflection points on
The duality of conformally flat manifolds
• Mathematics
• 2010
AbstractIn a joint work with Saji, the second and the third authors gave an intrinsic formulation of wave fronts and proved a realization theorem for wave fronts in space forms. As an application, we
Ak singlarities of wave fronts
• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 2009
Abstract In this paper, we discuss the recognition problem for Ak-type singularities on wave fronts. We give computable and simple criteria of these singularities, which will play a fundamental role
Extrinsic diameter of immersed flat tori in S3
• Mathematics
• 2011
Enomoto, Weiner and the first author showed the rigidity of the Clifford torus amongst the class of embedded flat tori in S3. In the proof of that result, an estimate of extrinsic diameter of flat
Horospherical flat surfaces in Hyperbolic 3-space
• Mathematics
• 2007
Recently we discovered a new geometry on submanifolds in hyperbolic $n$-space which is called {\it horospherical geometry}. Unfortunately this geometry is not invariant under the hyperbolic
A_2-singularities of hypersurfaces with non-negative sectional curvature in Euclidean space
• Mathematics
• 2010
In a previous work, the authors gave a definition of front bundles'. Using this, we give a realization theorem for wave fronts in space forms, like as in the fundamental theorem of surface theory.
A global theorem for singularities of maps between oriented 2-manifolds
Let M and N be smooth compact oriented connected 2-manifolds. Suppose/: M-*N is smooth and every pointp € M is either a fold point, cusp point, or regular point of / i.e., / is excellent in the sense
The mandala of Legendrian dualities for pseudo-spheres in Lorentz-Minkowski space and "flat" spacelike surfaces
• Mathematics
• 2009
Using the Legnedrian duarities between surfaces in pseudo-spheres in Lorentz-{Minkow}{ski} 4-space, we study various kind of flat surfaces in pseudo-spheres. We consider a surface in the