A unit vector field X on a Riemannian manifold determines a submanifold in the unit tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. It is known that the parallel fields are the trivial minima. In this paper, we obtain a lower bound for the volume in terms of the integrals of the 2i-symmetric functions of the second fundamental form of the orthogonal distribution to the field X. In the spheres , this lower bound is independent of X. Consequently, the volume of a unit vector field on an odd-sphere is always greater than the volume of the radial field. The main theorem on volumes is applied also to hyperbolic compact spaces, giving a non-trivial lower bound of the volume of unit fields.
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Fabiano B. Brito, Pablo M. Chacón, A. M. Naveira, On the volume of unit vector fields on spaces of constant sectional curvature. Comment. Math. Helv. 79 (2004), no. 2, pp. 300–316DOI 10.1007/S00014-004-0802-4