The purpose of this paper is to determine those homogeneous real hypersurfaces in a complex projective space Pn(C) of complex dimension n(^>2) which are orbits under analytic subgroups of the projective unitary group PU(n-\\-\\)> and to give some characterizations of those hypersurfaces. In § 1 from each effective Hermitian orthogonal symmetric Lie algebra of rank two we construct an example of homogeneous real hypersurface in Pn(C)y which we shall call a model space in Pn(C). In §2 we show that the class of all homogeneous real hypersurfaces in Pn{C) that are orbits under analytic subgroups of PU(n-\\-l) is exhausted by all model spaces. In §§3 and 4 we give some conditions for a real hypersurface in Pn(C) to be an orbit under an analytic subgroup of PU(n-\\-l) and in the course of proof we obtain a rigidity theorem in Pn(C) analogous to one for hypersurfaces in a real space form. The author would like to express his hearty thanks to Professor T. Takahashi for valuable discussions with him and his constant encouragement, and to Professor M. Takeuchi who made an original complicated proof of Lemma 2.3 short and clear.

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On homogeneous real hypersurfaces in a complex projective space

Let M be a real submanifold of CPm, and let J denote the complex structure. We begin by finding a formula for the location of the focal points of M in terms of its second fundamental form. This takes a particularly tractable form when M is a complex submanifold or a real hypersurface on which Ji is a principal vector for each unit normal ( to M. The rank of the focal map onto a sheet of the focal set of M is also computed in terms of the second fundamental form. In the case of a real hypersurface on which JE is principal with corresponding principal curvature ,u, if the map onto a sheet of the focal set corresponding to ,u has constant rank, then that sheet is a complex submanifold over which M is a tube of constant radius (Theorem 1). The other sheets of the focal set of such a hypersurface are given a real manifold structure in Theorem 2. These results are then employed as major tools in obtaining two classifications of real hypersurfaces in cPm. First, there are no totally umbilic real hypersurfaces in cPm, but we show: THEOREM 3. Let M be a connected real hypersurface in CPm, m > 3, with at most two distinct principal curvatures at each point. Then M is an open subset of a geodesic hypersphere. Secondly, we show that there are no Einstein real hypersurfaces in cPm and characterize the geodesic hyperspheres and two other classes of hypersurfaces in terms of a slightly less stringent requirement on the Ricci tensor in Theorem 4. One of the first results in the geometry of submanifolds is that an umbilic hypersurface M in Euclidean space must be an open subset of a hyperplane or sphere. The proof goes as follows: assume that the shape operator is a scalar multiple of the identity, A = AX, and use the Codazzi equation to show that X is constant. Then either X = 0, in which case M lies on a hyperplane, or the focal points fx(x) = x + (l/X)t, ( the unit normal, all coincide, and M lies on the sphere of radius 1 /X centered at the unique focal point. This simple idea suggests a plan of attack for classifying hypersurfaces in terms of the nature of the principal curvatures. Under fairly general conditions, the set of focal points corresponding to a principal curvature X can be given a differentiable Received by the editors December 19, 1980. Presented to the Society at its annual meeting in San Francisco, January 10, 1981. 1980 Mathematics Subject Classificatiom Primary 53B25, 53C40.

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Focal sets and real hypersurfaces in complex projective space

A principal circle bundle over a real hypersurface of a complex projective space CPn can be regarded as a hypersurface of an odddimensional sphere. From this standpoint we can establish a method to translate conditions imposed on a hypersurface of CPn into those imposed on a hypersurface of S2'+1. Some fundamental relations between the second fundamental tensor of a hypersurface of CPn and that of a hypersurface of S2n+1 are given. Introduction. As is well known a sphere S2n+1 of dimension 2n + 1 is a principal circle bundle over a complex projective space CPn and the Riemannian structure on CPn is given by the submersion ir: S2n+ 1 ~ CPn [4], [7]. This suggests that fundamental properties of a submersion would be applied to the study of real submanifolds of a complex projective space. In fact, H. B. Lawson [2] has made one step in this direction. His idea is to construct a principal circle bundle M2n over a real hypersurface M2n-1 of Cpn in such a way that M2n is a hypersurface of S2n + 1 and then to compare the length of the second fundamental tensors of M2n-1 and M2 n. Thus we can apply theorems on hypersurfaces of S2n+1. In this paper, using Lawson's method, we prove a theorem which characterizes some remarkable classes of real hypersurfaces of Cpn. First of all, in ?1, we state a lemma for a hypersurface of a Riemannian manifold of constant curvature for the later use. In ?2, we recall fundamental formulas of a submersion which are obtained in [4], [7] and those established between the second fundamental tensors of M and M. In ?3, we give some identities which are valid in a real hypersurface of CPn. After these preparations, we show, in ?4, a geometric meaning of the commutativity of the second fundamental tensor of M in Cp'n and a fundamental tensor of the submersion ir: M . M 1. Hypersurfaces of a Riemannian manifold of constant curvature. Let M be an (m + 1)-dimensional Riemannian manifold with a Riemannian metric G and i: M M be an isometric immersion of an m-dimensional differentiable Received by the editors March 25, 1974 and, in revised form, September 9, 1974. AMS (MOS) subject classifications (1970). Primary 53C40, 53C20.

On some real hypersurfaces of a complex projective space

The present volume is the written version of the series of lectures the author delivered at the Catholic University of Leuven, Belgium during the period of June-July, 1990. The main purpose of these talks is to present some of author's work and also his joint works with Professor T. Nagano and Professor Y. Tazawa of Japan, Professor P. F. Leung of Singapore and Professor J. M. Morvan of France on geometry of slant submanifolds and its related subjects in a systematical way.

Geometry of Slant Submanifolds

Rotation hypersurfaces in spaces of constant curvature are defined and their principal curvatures are computed. A local characterization of such hypersurfaces, with dimensions greater than two, is given in terms of principal curvatures. Some special cases of rotation hypersurfaces, with constant mean curvature, in hyperbolic space are studied. In particular, it is shown that the well-known conjugation between the helicoid and the catenoid in euclidean three-space extends naturally to hyperbolic three-space H3 ; in the latter case, catenoids are of three different types and the explicit correspondence is given. It is also shown that there exists a family of simply-connected, complete, embedded, nontotally geodesic stable minimal surfaces in H3.

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Rotation Hypersurfaces in Spaces of Constant Curvature

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Patrick Joseph Ryan

Papers

Homogeneity and some curvature conditions for hypersurfaces