Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
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COMPLEX SUBMANIFOLDS IN REAL HYPERSURFACES

  • Received : 2009.01.05
  • Published : 2010.09.01
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Abstract

Let M be a $C^{\infty}$ real hypersurface in $\mathbb{C}^{n+1}$, $n\;{\geq}\;1$, locally given as the zero locus of a $C^{\infty}$ real valued function r that is defined on a neighborhood of the reference point $P\;{\in}\;M$. For each k = 1,..., n we present a necessary and sufficient condition for there to exist a complex manifold of dimension k through P that is contained in M, assuming the Levi form has rank n - k at P. The problem is to find an integral manifold of the real 1-form $i{\partial}r$ on M whose tangent bundle is invariant under the complex structure tensor J. We present generalized versions of the Frobenius theorem and make use of them to prove the existence of complex submanifolds.

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Acknowledgement

Grant : Geometric Properties of Real and Complex Manifolds

Supported by : Korea Research Foundation, MURST

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Cited by

  1. Local Geometry of Levi-Forms Associated with the Existence of Complex Submanifolds and the Minimality of Generic CR Manifolds vol.22, pp.2, 2012, https://doi.org/10.1007/s12220-010-9208-2