Title: 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
$M^{2n+1}$, $n\geq 1$, 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 $M^{m}$,
$1\leq k\leq m$, 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 $\mathbb{S}^{2n+1}$.