  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 sphere and extend a result due to Shiohama and Xu \cite{SX} for compact
hypersurfaces in any space form.

Differential Geometry