The Boy-Gauss-Bonnet theorems for \(C^\infty\)-singular surfaces with limiting tangent bundle
The author investigates \(C^\infty\)-maps \(j\) of compact, oriented, smooth two dimensional manifolds \(M\) into the Euclidean three-space \(\mathbb{E}^3\) which are not immersions, i.e. where rank \(j_*<2\) for the critical locus \(D^0\subset M\). In order to get useful results he restricts himself to the case that there exists a \(C^\infty\)-global unit normal vector field of \(j(M)\). Further transversality assumptions are made for the points of \(D^0\) which imply that \(D^0\) is a disjoint union of \(C^\infty\)-embedded closed curves in \(M\). Then two global Gauss-Bonnet theorems are proved for the CW-complex \(j(M)\). Moreover applications for Weyl's tube formula and parallel surfaces are indicated.
scientific article; zbMATH DE number 1747786
scientific article