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For a Riemannian foliation on a closed manifold, the first secondaryinvariant of Molino's central sheaf is an obstruction to tautness. Anotherobstruction is the class defined by the basic component of the mean curvaturewith respect to some metric. Both obstructions are proved to be the same up toa constant, and other geometric properties are also proved to be equivalent totautness.

On the first secondary invariant of Molino's central sheaf