The Gauss–Bonnet Theorem for Coherent Tangent Bundles over Surfaces with Boundary and Its Applications

Abstract

In [34,35,36] the Gauss-Bonnet formulas for coherent tangent bundles over compact oriented surfaces (without boundary) were proved. We establish the Gauss-Bonnet theorem for coherent tangent bundles over compact oriented surfaces with boundary. We apply this theorem to investigate global properties of maps between surfaces with boundary. As a corollary of our results we obtain Fukuda-Ishikawa's theorem. We also study geometry of the affine extended wave fronts for planar closed non singular hedgehogs (rosettes). In particular, we find a link between the total geodesic curvature on the boundary and the total singular curvature of the affine extended wave front, which leads to a relation of integrals of functions of the width of a resette.2010 Mathematics Subject Classification. Primary 57R45, Secondary 53A05.

The geometry of the Wigner caustic and a decomposition of a curve into parallel arcs

“…Definition 2. 15 The tangent line of E 1 2 (M) at a cusp point p is the limit of a sequence of 1-dimensional vector spaces T q n M in RP 1 for any sequence q n of regular points of E 1 2 (M) converging to p. This definition does not depend on the choice of a converging sequence of regular points. By Lemma 2.7(i) we can see that the tangent line to E 1 2 (M) at the cusp point a+b 2 is parallel to tangent lines to M at a and b.…”

“…The geometry of an affine extended wave front, i.e. the set λ ∈ [0, 1]{λ} × E λ (M), where E λ (M) is an affine λ-equidistant of a manifold M, was studied in [11,15].…”

Singular Points of the Wigner Caustic and Affine Equidistants of Planar Curves

“…The geometry of an affine extended wave front, i.e. the set λ∈[0,1] {λ} × E λ (M ) was studied in [11]. Let p be a inflexion point of M .…”

Normal form of the swallowtail and its applications

“…A map f : M → R 3 between 2-dimensional closed manifold M and R 3 is called a frontal (respectively, a front) if there exists a map ν : M → S 2 such that for any p ∈ M, the map-germ f at p is a frontal (respectively, a front) whose unit normal is ν. We remark that ν is defined on M. Gauss-Bonnet type theorems for fronts are obtained in [19][20][21], and it is generalized to the case of ∂M = ∅ in [3], see also [9]. In these theorems, it is assumed that all singularities p of f satisfy rank df p = 1.…”

“…Wave fronts and frontals are surfaces with singularities in 3-space which have normal directions even along singularities. In these decades, there appeared several articles concerning on differential geometry of wave fronts and frontals [3,[6][7][8]10,[15][16][17][18]20]. Generic singularities of fronts in 3-space are cuspidal edges and swallowtails.…”