  Motivated by Gray's work on tube formulae for complex submanifolds of complex
projective space equipped with the Fubini-Study metric, Riemannian foliations
of projective space are studied. We prove that there are no complex Riemannian
foliations of any open subset of $\mathbb{P}^n$ of codimension one. As a
consequence there is no Riemannian foliation of the projective plane by Riemann
surfaces, even locally. We determine how a complex submanifold may arise as an
exceptional leaf of a non-trivial singular Riemannian foliation of maximal
dimension. Gray's tube formula is applied to obtain a volume bound for certain
holomorphic curves of complex quadrics.
