A topological lower bound for the energy of a unit vector field on a closed Euclidean hypersurface

Abstract

For a unit vector field on a closed immersed Euclidean hypersurface $M^{2n+1}$, $n\ge 1$, we exhibit a nontrivial lower bound for its energy which depends on the degree of the Gauss map of the immersion. When the hypersurface is the unit sphere ${\mathbb{S}}^{2n+1}$, immersed with degree one, this lower bound corresponds to a well established value from the literature. We introduce a list of functionals ${\mathcal{B}}_k$ on a compact Riemannian manifold $M^m$, $1\le k\le m$, and show that, when the underlying manifold is a closed hypersurface, these functionals possess similar properties regarding the degree of the immersion. In addition, we prove that Hopf flows minimize ${\mathcal{B}}_n$ on ${\mathbb{S}}^{2n+1}$.

12 pages. Extended and corrected version. Manuscript modified in order to emphasize some known results and the higher order total bending functionals