Almost conformally flat hypersurfaces

Abstract

We prove a universal lower bound for the L n/2 -norm of the Weyl tensor in terms of the Betti numbers for compact n-dimensional Riemannian manifolds that are conformally immersed as hypersurfaces in the Euclidean space. As a consequence, we determine the homology of almost conformally flat hypersurfaces. Furthermore, we provide a necessary condition for a compact Riemannian manifold to admit an isometric minimal immersion as a hypersurface in the round sphere and extend a result due to Shiohama and Xu [18] for compact hypersurfaces in any space form.

Topological obstructions for submanifolds in low codimension

“…Then it follows directly from (22) that n−k i=k τ i (f ) < 1. Thus, there exists u ∈ S n−k−1 such that the height function h u : M n → R is a Morse function whose number of critical points of index i satisfies µ i (u) = 0 for any k ≤ i ≤ n − k. The fundamental theorem of Morse theory (cf.…”

Obstruction to isometric immersions

“…In Section 3.4, we extend their result for compact hypersurfaces in spheres or in the hyperbolic space. This result is contained in [49].…”

“…Moreover, we obtain many applications: For instance, we provide a necessary condition for a compact Riemannian manifold to allow a conformal immersion as a hypersurface in the Euclidean space R n+1 . These results are contained in [49].…”