Differential Geometry of Complex Hypersurfaces

“…Parallel Kahler submanifolds of a complex projective space have been completely classified by Nakagawa and Takagi [8] as P", Q", Pk X P"-k, SU(n + 2)/S(U(2) X (/(«)) (n > 3), SO(\0)/U(5) and £6/Spin(10) X T\ All principal curvatures of a totally geodesic submanifold are zero, and Q" has two distinct principal curvatures for any normal directions [13]. Moreover, it is known by Ogiue [10] that principal curvatures of Veronese imbedding of the H-dimensional complex projective space of constant holomorphic sectional curvature 2 into P"<" + 3>/2 depends on the normal direction, hence it does not satisfy the conditions of Theorem 3.…”

Section: Propositionmentioning

confidence: 99%

Real hypersurfaces and complex submanifolds in complex projective space

Kimura¹

1986

Trans. Amer. Math. Soc.

236

117

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Abstract.Let M be a real hypersurface in /"'(C), J be the complex structure and | 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 Jt, is principal. We also obtain a characterization of certain complex submanifolds in a complex projective space. Specifically, /""(C) (totally geodesic), Q", Pl(C) x P"(C). SU{5)/S(U{2) X (7(3)) and SO(10)/t/(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. 0. Introduction. Many differential geometers have investigated isoparametric hypersurfaces in a sphere (i.e., submanifolds of codimension 1 with constant principal curvatures). In particular, Münzner [6, 7] showed that the number of distinct principal curvatures of isoparametric hypersurfaces in a sphere is 1,2,3,4 or 6. This result is not obtained by classfying all isoparametric hypersursfaces.There are many papers about real hypersurfaces in complex projective space P"(C). In particular, Takagi [15] classfied all homogeneous real hypersurfaces in P"(C), and showed that the number of distinct principal curvatures of homogeneous real hypersurfaces in P"(C) is 2,3 or 5. Moreover, if a real hypersurface M has 2 or 3 distinct constant principal curvatures, then M is congruent to one of the homogeneous examples [16 and 17].In Theorem 1, using the results of Münzner and Okumura, we show that if a real hypersurface M of P"(C) has constant principal curvatures and if the tangent vector field ./£, which is obtained by applying the complex structure J to the unit normal vector field |, is principal, then the number of distinct principal curvatures of M is 2, 3 or 5. Cecil and Ryan [4] studied real hypersurfaces in P"(C) on which J£ is principal. In fact, the tangent vector field /£ on a tube over a Kahler submanifold is principal. Conversely, if J£ is principal on a real hypersurface and the rank of focal map is constant ( §2), then M lies on a tube over a Kahler submanifold. Making use of these these results, we have Theorem 4. Let M be a connected real hypersurface in P"(C). Then M has constant principal curvatures and Ji-is principal if and only if M is congruent to an open subset of a homogeneous real hypersurface.

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“…Among the other different type of Hermitian symmetric space with rank 2 in the class of compact type, we can give the example of complex quadric Q m = SO m+2 /SO m SO 2 , which is a complex hypersurface in complex projective space CP m+1 (see Suh [16], [18], and Smyth [11]). The complex quadric can also be regarded as a kind of real Grassmann manifolds of compact type with rank 2 (see Kobayashi and Nomizu [5]).…”

Section: Introductionmentioning

confidence: 99%