Riemannian foliations of projective space admitting complex leaves

2012/02/27 by Thomas Murphy, Murphy, Thomas
Mathematics · #53C12 #53C40 #Differential Geometry (math.DG) #FOS: Mathematics #math.DG #msc:53C12 #msc:53C40

paper · pdf · doi:10.48550/arxiv.1202.5989

To appear in Rend. Semin. Mat. Univ. Politec. Torino

arxiv created 2013/07/10 · arxiv updated 2013/07/11

Abstract

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 ℙⁿ 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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