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

摘要

For a unit vector field on a closed immersed Euclidean hypersurface M2n+1, n >= 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 S2n+1, immersed with degree 1, this lower bound corresponds to a well established value from the literature. We introduce a list of functionals B-k on a compact Riemannian manifold M-m, 1 <= k <= 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 B-n on S2n+1.