Abstract
A vector field X on a Riemannian manifold determines a submanifold in the tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. When M is compact, the volume is well defined and, usually, this functional is studied for unit fields. Parallel vector fields are trivial minima of this functional. For manifolds of dimension 5, we obtain an explicit result showing how the topology of a vector field with constant length influences its volume. We apply this result to the case of vector fields that define Riemannian foliations with all leaves compact.
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