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Transactions of the American Mathematical Society

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Real hypersurfaces and complex submanifolds in complex projective space


Author: Makoto Kimura
Journal: Trans. Amer. Math. Soc. 296 (1986), 137-149
MSC: Primary 53C40
DOI: https://doi.org/10.1090/S0002-9947-1986-0837803-2
MathSciNet review: 837803
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Abstract: Let $ M$ be a real hypersurface in $ {P^n}({\mathbf{C}})$ be the complex structure and $ \xi $ denote a unit normal vector field on $ M$. We show that $ M$ is (an open subset of) a homogeneous hypersurface if and only if $ M$ has constant principal curvatures and $ J\xi $ is principal. We also obtain a characterization of certain complex submanifolds in a complex projective space. Specifically, $ {P^m}({\mathbf{C}})$ (totally geodesic), $ {Q^n},{P^1}({\mathbf{C}}) \times {P^n}({\mathbf{C}}),SU(5)/S(U(2) \times U(3))$ and $ SO(10)/U(5)$ are the only complex submanifolds whose principal curvatures are constant in the sense that they depend neither on the point of the submanifold nor on the normal vector.


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DOI: https://doi.org/10.1090/S0002-9947-1986-0837803-2
Keywords: Real hypersurfaces, complex submanifolds, focal sets, tubes, shape operators
Article copyright: © Copyright 1986 American Mathematical Society

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