Curvature locus of a 3-manifold having a normal bundle with some flat direction

Abstract

AbstractA well known result states that the curvature of the normal bundle of ann-manifold immersed in the Euclidean space,$${\mathbb {R}}^{n+k}$$Rn+k, vanishes if and only if the curvature locus at any point is a convex polytope. In the case of 3-manifolds, the curvature locus is, generically, a triangle. In this paper we determine, among all the possible curvature locii of 3-manifolds immersed in$${\mathbb {R}}^{3+3}$$R3+3, which ones can be projected on a triangle and we relate them with the curvature of the normal bundle.