## CR submanifolds of complex projective space.(English)Zbl 1187.32031

Developments in Mathematics 19. Berlin: Springer (ISBN 978-1-4419-0433-1/hbk; 978-1-4419-0434-8/ebook). viii, 176 p. (2010).
This text deals with $$CR$$ submanifolds of complex manifolds, with special emphasis on $$CR$$ submanifolds of complex projective spaces.
The first part of the book is addressed to graduate students, having only knowledge of introductory manifold theory and curvature properties of Riemannian geometry. Thus, the first chapters contain basic introductory material, including almost complex structures, Kähler manifolds, submersions and immersions, the structure equations of submanifolds, the shape operator, the Levi form of $$CR$$ submanifolds, and topics like the study of special submanifolds of the real sphere, codimension reduction, principal circle bundles. This part is based on lectures at Saitama University, Japan, given by the second author.
The second part of the text is devoted to real hypersurfaces and $$CR$$ submanifolds of complex manifolds, with particular emphasis on $$CR$$ submanifold of complex projective spaces having maximal $$CR$$ dimension. Some important result, that have been recently published on mathematical journal, are here presented for the first time in the book form. The last six chapters contain original results by the Authors with complete proofs. These topics include conditions for the existence of Levi-flat hypersurfaces, embedding into geodesic hyperspheres in the projective space, differential-geometric characterization of some special invariant submanifolds of maximal $$CR$$ dimension of complex space forms.

### MSC:

 32V40 Real submanifolds in complex manifolds 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 32-02 Research exposition (monographs, survey articles) pertaining to several complex variables and analytic spaces 32-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to several complex variables and analytic spaces 32Q60 Almost complex manifolds 53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)

### Keywords:

complex projective space; $$CR$$ submanifold
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