Real hypersurfaces and complex submanifolds in complex projective space

Abstract

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.

Real Hypersurfaces With Cyclic-Parallel Structure Jacobi Operators in a Nonflat Complex Space Form

“…By the classification theorems of real hypersurfaces in M n (c), c = 0, due to Kimura [7] and Berndt [1], M is locally congruent to one of the homogeneous real hypersurfaces of type A 1 ∼ E in P n C or of type A 0 ∼ B in H n C. So, we shall check equation (3.2) one by one for the above model spaces.…”

Complete real hypersurfaces in compact rank one symmetric spaces

“…Berndt [1] noticed that the constancy of the principal curvatures of M ⊂ QP n (c) is not necessary. There are a lot of results on the local shapes of such hypersurfaces, for instance [4] and others. The point is whether a certain topological restriction is imposed if completeness is assumed for M .…”

Real hypersurfaces in the complex quadric with commuting Ricci tensor

“…In the complex projective space CP m+1 and the quaternionic projective space QP m+1 some classifications related to commuting Ricci tensor or commuting structure Jacobi operator were investigated by Kimura [3], [4], Pérez [6] and Pérez and Suh [7], [8] respectively. Under the invariance of the shape operator along some distributions a new classification in the complex 2-plane Grassmannian G 2 (C m+2 ) = SU m+2 /S(U m U 2 ) was investigated.…”