Real hypersurfaces and complex submanifolds in complex projective space

Kimura, Makoto

doi:10.1090/S0002-9947-1986-0837803-2

Real hypersurfaces and complex submanifolds in complex projective space
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Trans. Amer. Math. Soc. 296 (1986), 137-149 Request permission

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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