Total Curvature for C∞-Singular Surfaces with Limiting Tangent Bundle

Abstract

The total curvature of a compact C∞-immersed surface in Euclidean 3-space \({\cal E}\)3 can be interpreted as the average number of critical points for a linear ‘height’ function. The Morse inequalities provide an intrinsic topological lower bound for the total curvature and ‘tight’ surfaces, which realize equality, have been an active topic of research. The objective of this paper is to describe the natural notion of total curvature for C∞-singular surfaces which fail to immerse on C∞-embedded closed curves, but which have a C∞-globally defined unit normal (e.g. caustics, or critical images for mappings of 3-manifolds into Euclidean 3-space). For such surfaces total curvature consists of a sum of two-dimensional and one-dimensional integrals, which have various lower bounds. Large sets of LT-surfaces which realize equality are then constructed. As an application, the orthogonal projection of an immersed tight hypersurface in Euclidean 4-space is shown to have LT-tight critical image, and several related inequalities are given.