Ricci tensors of real hypersurfaces in a complex projective space

Abstract

Abstract. This paper gives a classification of real hypersurfaces in a complex projective space under assumptions that the structure vector £ is principal, the focal map has constant rank, and V¿S = 0, where S is the Ricci tensor of the real hypersurface.

Real hypersurfaces of a complex space form

“…(2.8), we get that these distributions are integrable with integral submanifolds of D 1 totally geodesic submanifolds of (C n+1 2 , J, , ) and those of D 2 totally umbilical submanifolds of (C n+1 2 , J, , ), we get that M is locally congruent to R n−k × S n (c). [12][13][14][15][16][17][18]). Recall that a smooth vector field u on a complex manifold (M, J ) with complex structure J is said to be an analytic vector field if the local flow {ψ t } of u consists of almost complex local diffeomorphisms ψ t of M, that is the differential dψ t of ψ t commutes with the complex structure J .…”

“…For instance M(c), c = 0 does not admit totally umbilical or Einstein real hypersurfaces as well as their geodesic spheres do not have constant curvature. Compact minimal real hypersurfaces, real hypersurfaces of constant mean curvature and Hopf hypersurfaces in a complex space form M(c) and in nearly Kaehler sphere S 6 are studied in [1,[7][8][9][11][12][13][16][17][18][20][21][22]. In general, the study of real hypersurfaces in a complex space form M(c), c = 0 is a difficult area as compared to the study of hypersurfaces in real space forms (Riemannian manifolds of constant sectional curvature).…”

Ricci tensor of Hopf hypersurfaces in a complex space form

“…Suh and Maeda classified Hopf hypersurfaces of M n (4c) with η-parallel Ricci tensor ( [11], [9]). In [8], Maeda gave a classification of Hopf hypersurfaces in CP n with ∇ ξ S = 0.…”

Real hypersurfaces of a complex projective space

“…For instanceM(c), c = 0, does not admit totally umbilical or Einstein hypersurfaces as well as their geodesic spheres do not have constant curvature. Compact minimal real hypersurfaces, real hypersurfaces of constant mean curvature and Hopf hypersurfaces in a complex space form and in nearly Kaehler sphere S 6 are studied in [1,4,[6][7][8][9][10][11][12][13][15][16][17] and these hypersurfaces are fully characterized up to some dimensional constraints. Homogeneous real hypersurfaces in a complex space form are completely classified (cf.…”