Riemannian foliations of projective space admitting complex leaves

Details Riemannian foliations of projective space admitting complex leaves Riemannian foliations of projective space admitting complex leaves

2012-02-27

arxiv.org/abs/1202.5989v1

Mathematics

Differential Geometry

Scientific paper

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 locally complex Riemannian foliations 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.

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