The total absolute curvature of submanifolds with singularities

In this paper, we give a generalization of the Chern-Lashof theorem for submanifolds with singularities called frontals in Euclidean space. We prove that, for an $n$-dimensional admissible compact frontal in $(n+r)$-dimensional Euclidean space ${\mathit{R}}^{n+r}$, its total absolute curvature is greater than or equal to the sum of the Betti numbers. Furthermore, if the total absolute curvature is equal to $2$, and all singularities are of the first kind, then the image of the frontal coincides with a closed convex domain of an affine $n$-dimensional subspace of ${\mathit{R}}^{n+r}$.