Characterization of totally umbilical hypersurfaces

The theory of transnormality has arisen from attempts to generalize a concept of curves with constant width in a Euclidian plane to an analogue in a Riemannian manifold. S.A. Robertson [4, 5] has accomplished this in the case where the ambient space is Euclidian. Subsequently, J. Bolton [1] has extended this to the case of a hypersurface in a Riemannian manifold. Let M be an w-dimensional connected complete Riemannian manifold isometricallyimbedded into an (w+l)-dimensional connected complete Riemannian manifold M. For each xgM there exists,up to parametrization, a unique geodesic rx of M which intersects M orthogonally at x. M is called a transnormal hypersurface of M if, for each pair x,y£M, the relation rx3y implies that rx ―xv. We difine an equivalent relation ~ for points on a transnormal hypersurface M by writing x~y to mean ysrx, with respect to which the quotient space M=M/~ with the quotient topology is considered. M is called an r-transnormal hypersurface if the natural projection <p of M onto M is an r-fold covering map. Recently, S. Nishikawa [2] has studied some global properties of transnormal hypersurfaces of a complete Riemannian manifold. He gave also in [3] differential geometric structures of a compact 2-transnormal hypersurface of a simply connected complete Riemannian manifold of constant curvature. In this paper, we shall investigate 2-transnormal hypersurfaces in a Kaehler manifold of constant holomorphic sectional curvature and prove that these hypersurfaces are geodesic hyperspheres if principal curvatures are bounded from below (or above) by certain constants which depend only upon the diameters of the hypersurfaces and the holomorphic sectional curvatures of the ambient Kaehler manifolds. (Theorem 3.3 and Theorem 4.1) The auther would like to express her hearty gratitude to Prof. Shun-ichi Tachibana for his kind advice and suggestion.

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.

where Rijk denotes the covariant derivative of Ricci tensor Rfy. This conditionis essentially weaker than that for the parallel Recci tensor. In fact Derdzinski [2] gave an example of a 4-dimentional Riemannian manifold with harmonic curvaiure whose Ricci tensor is not parallel. Recently E. Omachi [5] investigated compact hypersurfaces with harmonic curvature in a Euclidean space or a sphere and gave a classification of such hypersurfaces provided that the mean curvature is constant. This paper is concerned with hypersurfaces with harmonic curvature isometrically immersed into a Riemannian manifold of constant curvature. In the firstsection,a concept of Codazzi type for a symmetric (0, 2)-tensor is introduced and a sufficientcondition for a symmetric tensor of Codazzi type to be parallelis given. A similar condition for a symmetric tensor of Codazzi type is also treated by S. Y. Cheng an S. T. Yau [1]. In the second section, the result proved in the first section is applied to hypersurfaces with harmonic curvature immersed in a Riemannian manifold of constant curvature, in which Omachi's result [5] is generalized without the assumption of compactness. Finally we study also the case where the assumotion that the mean curvature is constant is omitted.

where Ai(p) and A2(p) are the principal curvatures of M at p. When H is constant, M is called a surface of constant mean curvature. A surface is said to have finite type if it is homeomorphic to a closed surface with a finite number of points removed. An important problem in classical differential geometry is the classification of properly embedded finite type surfaces M of constant mean curvature in R . If M is a closed embedded surface of constant mean curvature, then it follows from Alexandrov [1] that M must be a round sphere. The classical examples of properly embedded surfaces with zero mean curvature are the plane, the helicoid and the catenoid. Surfaces of zero mean curvature are usually called minimal surfaces. The remaining classical examples of properly embedded surfaces of constant mean curvature were found by Delaunay [4]. The Delaunay surfaces are surfaces of revolution. Recently Hoffman and Meeks [6, 7] have found examples of properly embedded minimal surfaces which are homeomorphic to closed surfaces of positive genus with 3 points removed. Callahan, Hoffman and Meeks [3] have found other examples with more ends. An annular end E of & properly embedded surface in R 3 is a properly embedded annulus E in M where E is homeomorphic to S x [0,1). When M has finite type, then every end of M is annular. Hoffman and Meeks have developed a theory to deal with global problems concerning the geometry of properly embedded minimal surfaces M and, in particular, they show that most annular ends of M converge at infinity in R to a flat plane or to the end of a catenoid. Recently N. Kapouleas [8] in his thesis has shown that for every positive integer k > 2, there exists a properly embedded surface M& of finite type with nonzero mean curvature and with k ends. He also has constructed highergenus examples. As in the case of minimal surfaces, the annular ends of a properly embedded surface of nonzero constant mean curvature have a special geometry and play an important role in global theorems.

This paper gives a sufficient condition for a complete hypersurface of a Riemannian manifold of constant curvature to be umbilical. The condition will be given by an inequality which is established between the length of the second fundamental tensor and the mean curvature.

In our paper we have determined the dimension of the space of conformal Killing-Yano tensors and the dimensions of its two subspaces of closed conformal Killing-Yano and Killing-Yano tensors on pseudo Riemannian manifolds of constant curvature. This result is a generalization of well known results on sharp upper bounds of the dimensions of the vector spaces of conformal Killing-Yano, Killing-Yano and concircular vector fields on pseudo Riemannian manifolds of constant curvature.

A strictly pseudoconvex pseudo-Hermitian manifoldM admits a canonical Lorentz metric as well as a canonical Riemannian metric. Using these metrics, we can define a curvaturelike function Λ onM. AsM supports a contact form, there exists a characteristic vector field ξ dual to the contact structure. If ξ induces a local one-parameter group ofCR transformations, then a strictly pseudoconvex pseudo-Hermitian manifoldM is said to be a standard pseudo-Hermitian manifold. We study topological and geometric properties of standard pseudo-Hermitian manifolds of positive curvature Λ or of nonpositive curvature Λ. By the definition, standard pseudo-Hermitian manifolds are calledK-contact manifolds by Sasaki. In particular, standard pseudo-Hermitian manifolds of constant curvature Λ turn out to be Sasakian space forms. It is well known that a conformally flat manifold contains a class of Riemannian manifolds of constant curvature. A sphericalCR manifold is aCR manifold whose Chern-Moser curvature form vanishes (equivalently, Weyl pseudo-conformal curvature tensor vanishes). In contrast, it is emphasized that a sphericalCR manifold contains a class of standard pseudo-Hermitian manifolds of constant curvature Λ (i.e., Sasakian space forms). We shall classify those compact Sasakian space forms. When Λ≤0, standard pseudo-Hermitian closed aspherical manifolds are shown to be Seifert fiber spaces. We consider a deformation of standard pseudo-Hermitian structure preserving a sphericalCR structure.

Homogeneous graded metrics over split ℤ2-graded manifolds whose Levi-Civita connection is adapted to a given splitting, in the sense recently introduced by Koszul, are completely described. A subclass of such is singled out by the vanishing of certain components of the graded curvature tensor, a condition that plays a role similar to the closedness of a graded symplectic form in graded symplectic geometry: It amounts to determining a graded metric by the data {g, ω, Δ′}, whereg is a metric tensor onM, ω 0 is a fibered nondegenerate skewsymmetric bilinear form on the Batchelor bundleE → M, and Δ′ is a connection onE satisfying Δ′ω = 0. Odd metrics are also studied under the same criterion and they are specified by the data {κ, Δ′}, with κ ∈ Hom (TM, E) invertible, and Δ′κ = 0. It is shown in general that even graded metrics of constant graded curvature can be supported only over a Riemannian manifold of constant curvature, and the curvature of Δ′ onE satisfiesRΔ′ (X,Y)2 = 0. It is shown that graded Ricci flat even metrics are supported over Ricci flat manifolds and the curvature of the connection Δ′ satisfies a specific set of equations. 0 Finally, graded Einstein even metrics can be supported only over Ricci flat Riemannian manifolds. Related results for graded metrics on Ω(M) are also discussed.

This note describes an observation connecting Riemannian manifolds of constant sectional curvature with a particular class of Lie superalgebras. Specifically, it is shown that the structural equations of a space M with constant sectional curvature, of one variety or another, nearly coincide with some identities satisfied by tensors which can be used to construct some specific families of Lie superalgebras. In particular, one obtains either osp(n,2), spl(n,2), or osp(4,2n) if the Riemannian manifold has constant curvature, constant holomorphic curvature or constant quaternion-holomorphic curvature, respectively.

tures and n respectively, dim V1 = m> 1 and dim V2 =n-m ?1. Mr n-mn In the latter part of this statement, the second fundamental form A has two eigenvalues of multiplicities dim VI and dim V2. The main purpose of the present paper is to investigate the converse problem for minimal hypersurfaces in Sn+1. The author will prove a local theorem on the integrability of the distributions of the spaces of principal vectors of a hypersurface in a Riemannian manifold of constant curvature (Theorem 2). Making use of it, he will investigate minimal hypersurfaces such that the multiplicities of principal curvatures are constant. He will prove that if the number of principal curvatures is two and the multiplicities of them are at least two for a minimal hypersurface of this kind in Sn+', then it is

The theory of hypersurfaces, defined as submanifolds of codimension one, is one of the most fundamental theories of submanifolds. Therefore, in Sections 11–13 we consider hypersurfaces of a Riemannian manifold of constant curvature. This research, combined with the results obtained in Section 10, will contribute to studying real hypersurfaces of complex projective space in Section 16.

One of the most important results in differential geometry is that the only closed hypersurfaces of constant mean curvature and in general constant higher order mean curvature) embedded in Euclidean space are round spheres [1]. This result is not true for the case of immersed (and non-embedded hypersurfaces [11, 14]. Many generalizations of this result have been obtained later, for example constant scalar curvatures or constant higher order mean curvatures hypersurfaces [2,3,7,9]. As a natural generalization of hypersurfaces with constant mean curvature or with constant higher order mean curvature, linear Weingarten and more general generalized, Weingarten hypersurfaces hypersurface has been studied in many places. [5],[10]. The aim of our work is to establish a characterization theorem concerning complete generalized Weingarten hypersurfaces embedded in Euclidean space. That is an hypersurfaces where some of the higher order mean curvature are linearly related. We prove that the only closed, oriented generalized Weingarten hypersurfaces embedded in Euclidean space with non-vanishing higher order mean curvature are the round spheres. This result generalizes the cases of constant higher order mean curvature hypersurfaces and linear Weingarten hypersurfaces embedded in Euclidean space.

The mathematical formalism necessary for the diagrammatic evaluation of quantum corrections to a conformally invariant field theory for a self-interacting scalar field on a curved manifold with boundary is considered. The evaluation of quantum corrections to the effective action past one-loop necessitates diagrammatic techniques. Diagrammatic evaluations and higher loop-order renormalization can be best accomplished on a Riemannian manifold of constant curvature accommodating a boundary of constant extrinsic curvature. In such a context, the stated evaluations can be accomplished through a consistent interpretation of the Feynman rules within the spherical formulation of the theory which the method of images allows. To this effect, the mathematical consequences of such an interpretation are analysed and the spherical formulation of the Feynman rules on the bounded manifold is, as a result, developed.

Constant mean curvature hypersurfaces constitute a very important class of submanifolds in a compact Riemannian manifold (M, g). In this paper we are interested in families of such submanifolds, with mean curvature varying from one member of the family to another, which form (partial) foliations and which ‘condense’ to a submanifold Γ ⊂ M of codimension greater than 1. Our main results concern the existence of such families and, conversely, the geometric nature of the submanifolds Γ to which such families can condense. The simplest case, where Γ is a point, was considered by Ye a decade ago, [12], [13]. He proved that if p ∈ M is a nondegenerate critical point of the scalar curvature function Rg, then there exists a neighborhood U 3 p such that U \{p} is foliated by constant mean curvature (for short CMC) spheres; in fact, the members of this family are small perturbations of the geodesic spheres of radius ρ, 0 < ρ < ρ0, and hence they have mean curvatures H = 1/ρ → ∞. Moreover, this foliation is essentially unique. Conversely, if a neighbourhood of p admits such a foliation, then necessarily ∇Rg|p = 0. In very closely related work, Ye [14], and by quite different methods (using inverse mean curvature flow) Huisken and Yau [4], proved the existence of a unique foliation by CMC spheres near infinity in an asymptotically flat manifold (of nonnegative scalar curvature); this is of interest in general relativity. In this paper we study the existence of families of CMC hypersurfaces which converge to a (closed, embedded) submanifold Γ ⊂ M, particularly in the case ` = 1. Define the geodesic tube

On hypersurfaces with constant scalar curvature in a Riemannian manifold of constant curvature