The geometry and the Laplace operator on the exterior 2-forms on a compact Riemannian manifold

Let EF be a foliation on a Riemannian manifold M. The distribution on M which is defined to be orthogonal to SF is said to be normal to £Fand denoted by 3L. Nakagawa and Takagi [8] showed that any harmonic foliation on a compact Riemannian manifold of non-negative constant sectional curvature is totally geodesic if the normal distribution is minnimal. And succesively the present author [2] proved a complex version of their result, that is, the above result holds also on a complex projective space with a Fubini-Study metric. However, recently, Li [4] pointed out a serious mistake in the proof of the result of Nakagawa and Takagi, and so of the author's. Therefore those results are now open yet. On the other hand, Li [4] have studied a harmonic foliationon the sphere along the method of Nakagawa and Takagi, and obtained some interesting results. The purpose of this paper is to give a complex analogue of the Li's results. Let Pn+p(C) be the complex projective space with the Fubini-Study metric of constant holomorphic sectional curvature c. Let £Fbe a complex foliation on Pn+p(C) with ^-complex codimention and h the second fundamental tensor of <7. Then we shall Drove the following:

We study harmonic morphisms by placing them into the context of conformal foliations. Most of the results we obtain hold for fibres of dimension one and codomains of dimension not equal to two. We consider foliations which produce harmonic morphisms on both compact and noncompact Riemannian manifolds. By using integral formulae, we prove an extension to one-dimensional foliations which produce harmonic morphisms of the well-known result of S. Bochner concerning Killing fields on compact Riemannian manifolds with nonpositive Ricci curvature. From the noncompact case, we improve a result of R. L. Bryant[9] regarding harmonic morphisms with one-dimensional fibres defined on Riemannian manifolds of dimension at least four with constant sectional curvature. Our method gives an entirely new and geometrical proof of Bryant's result. The concept of homothetic foliation (or, more generally, homothetic distribution) which we introduce, appears as a useful tool both in proofs and in providing new examples of harmonic morphisms, with fibres of any dimension.

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. K. Nomizu and B. Smyth in [3] established a formula for the Laplacian of the second fundamental form of a hypersurface M immersed with constant mean curvature in a space M of constant sectional curvature c. Later, M. Okumura in [4] characterized under certain conditions a totally umbilical hypersurface of a Riemannian manifold of nonnegative constant curvature by an inequality between the length of the second fundamental tensor and the mean curvature of the hypersurface. In the present article we prove the following theorem. THEOREM A. Let M be an n-dimensional (n > 3) connected complete hypersurface immersed with constant mean curvature in an (n + 1)-dimensional Riemannian manifold M of positive constant curvature c. If the second fundamental tensor L satisfies trace L2 cn or M is totally geodesic. 1. Preliminaries. Let M be an (n + 1)-dimensional Riemannian manifold of constant curvature c. Let q: M -, M be an isometric immersion of an n-dimensional manifold M into M. In what follows we identify M with p(M) and p E M with (p(p) E rp(M) c M. The tangent space TpM is also identified with a subspace of T9,(p)M. The Riemannian metric g of M is induced from the Riemannian metric Received by the editors December 27, 1979 and, in revised form, February 26, 1980. AMS (MOS) subject classifications (1970). Primary 53C40, 53C20.

Let $\mathcal{C}(\mathcal{R},n,p,\Lambda,D,V_0)$ be the class of compact $n$-dimensional Riemannian manifolds with finite diameter $\leq D$, non-collapsing volume $\geq V_0$ and $L^p$-bounded $\mathcal{R}$-curvature condition $\|\mathcal{R}\|_{L^p}\leq \Lambda$ for some $p>\frac n2$. Let $(M,g_0)$ be a compact Riemannian manifold and $\mathcal{C}(M,g_0)$ the class of manifolds $(M,g)$ conformal to $(M,g_0)$. In this paper we use $\varepsilon$-regularity to show a rigidity result in the conformal class $\mathcal{C}(S^n,g_0)$ of standard sphere under $L^p$-scalar rigidity condition. Then we use harmonic coordinate to show $C^{\alpha}$-compactness of the class $\mathcal{C}(K,n,p,\Lambda,D,V_0)$ with additional positive Yamabe constant condition, where $K$ is the sectional curvature, and this result will imply a generalization of Mumford's lemma. Combining these methods together we give a geometric proof of $C^{\alpha}$-compactness of the class $\mathcal{C}(K,n,p,\Lambda,D,V_0)\cap \mathcal{C}(M,g_0)$. By using Weyl tensor and a blow down argument, we can replace the sectional curvature condition by Ricci curvature and get our main result that the class $\mathcal{C}(Ric,n,p,\Lambda,D,V_0)\cap \mathcal{C}(M,g_0)$ has $C^{\alpha}$-compactness.

The goal of this paper is to prove new upper bounds for the first positive eigenvalue of the [Formula: see text]-Laplacian operator in terms of the mean curvature and constant sectional curvature on Riemannian manifolds. In particular, we provide various estimates of the first eigenvalue of the [Formula: see text]-Laplacian operator on closed orientate [Formula: see text]-dimensional Lagrangian submanifolds in a complex space form [Formula: see text] with constant holomorphic sectional curvature [Formula: see text]. As applications of our main theorem, we generalize the Reilly-inequality for the Laplacian [R. C. Reilly, On the first eigenvalue of the Laplacian for compact submanifolds of Euclidean space, Comment. Math. Helv. 52(4) (1977) 525–533] to the [Formula: see text]-Laplacian for a Lagrangian submanifold in a complex Euclidean space and complex projective space for [Formula: see text] and [Formula: see text], respectively.

It is well known that the spectrum of Laplacian on a compact Riemannian manifold M is an important analytic invariant and has important geometric meanings. There are many mathematicians to investigate properties of the spectrum of Laplacian and to estimate the spectrum in term of the other geometric quantities of M . When M is a bounded domain in Euclidean spaces, a compact homogeneous Riemannian manifold, a bounded domain in the standard unit sphere or a compact minimal submanifold in the standard unit sphere, the estimates of the k + 1 -th eigenvalue were given by the first k eigenvalues (see [9], [12], [19], [20], [22], [23], [24] and [25]). In this paper, we shall consider the eigenvalue problem of the Laplacian on compact Riemannian manifolds. First of all, we shall give a general inequality of eigenvalues. As its applications, we study the eigenvalue problem of the Laplacian on a bounded domain in the standard complex projective space C P n ( 4 ) and on a compact complex hypersurface without boundary in C P n ( 4 ) . We shall give an explicit estimate of the k + 1 -th eigenvalue of Laplacian on such objects by its first k eigenvalues.

The purpose of this paper is to give a rigidity theorem for real hypersurfaces in Pn(C) satisfying a certain geometric condition. Introduction. Let Pn{C) denote an n{ > 2)-dimensional complex projective space with the metric of constant holomorphic sectional curvature 4c. We proved in [4] that two isometric immersions of a {In — l)-dimensional Riemannian manifold M into Pn(C) are congruent if their second fundamental forms coincide. In general, the type number is defined as the rank of the second fundamental form. In this paper we shall give another rigidity theorem of the same type: THEOREM A. Let M be a {In — \)-dimensίonal Riemannian manifold, and i and ϊ be two isometric immersions of M into Pn{C) {n > 3). Assume that i and ΐ have a principal direction in common at each point of M, and that the type number of (M, ί) or (M, ΐ) is not equal to 2 at each point of M. Then i and i are congruent, that is, there is a unique isometry φ of Pn{C) such that φoi = i. We shall say that an isometry φ of a real hypersurface M in Pn{C) is principal if for each point p of M there exists a principal vector v at p such that the vector φ*{v) is also principal at φ{p), where φ^ denotes the differential of φ at p. Then as an application of Theorem A we have: THEOREM B. Let M be a homogeneous real hypersurface in Pn{C) {n>3). Assume that each isometry of M is principal. Then M is an orbit under an analytic subgroup of the projective unitary group PU{n+ 1). Note that all orbits in Pn{C) under analytic subgroups of the projective unitary group PU{n+ 1) are completely classified in [4]. The authors would like to express their thanks to the referee for his useful advice. 1. Preliminaries. Let M be a {2n— l)-dimensional Riemannian manifold, and i be 1980 Mathematics Subject Classification (1985 Revision). Primary 53C40; Secondary 53C15.

The purpose of this paper is to give a rigidity theorem for real hypersurfaces in Pn(C) satisfying a certain geometric condition. Introduction. Let Pn{C) denote an n{ > 2)-dimensional complex projective space with the metric of constant holomorphic sectional curvature 4c. We proved in [4] that two isometric immersions of a {In — l)-dimensional Riemannian manifold M into Pn(C) are congruent if their second fundamental forms coincide. In general, the type number is defined as the rank of the second fundamental form. In this paper we shall give another rigidity theorem of the same type: THEOREM A. Let M be a {In — \)-dimensίonal Riemannian manifold, and i and ϊ be two isometric immersions of M into Pn{C) {n > 3). Assume that i and ΐ have a principal direction in common at each point of M, and that the type number of (M, ί) or (M, ΐ) is not equal to 2 at each point of M. Then i and i are congruent, that is, there is a unique isometry φ of Pn{C) such that φoi = i. We shall say that an isometry φ of a real hypersurface M in Pn{C) is principal if for each point p of M there exists a principal vector v at p such that the vector φ*{v) is also principal at φ{p), where φ^ denotes the differential of φ at p. Then as an application of Theorem A we have: THEOREM B. Let M be a homogeneous real hypersurface in Pn{C) {n>3). Assume that each isometry of M is principal. Then M is an orbit under an analytic subgroup of the projective unitary group PU{n+ 1). Note that all orbits in Pn{C) under analytic subgroups of the projective unitary group PU{n+ 1) are completely classified in [4]. The authors would like to express their thanks to the referee for his useful advice. 1. Preliminaries. Let M be a {2n— l)-dimensional Riemannian manifold, and i be 1980 Mathematics Subject Classification (1985 Revision). Primary 53C40; Secondary 53C15.

The natural symmetries of Riemannian manifolds are described by the symmetries of its Riemann curvature tensor. In that sense, the most symmetric manifolds are the constant sectional curvature ones. Its natural generalizations are locally symmetric manifolds, semisymmetric manifolds, and pseudosymmetric manifolds. The analogous generalizations of constant holomorphic sectional curvature Kaehler manifolds are locally symmetric Kaehler manifolds, semisymmetric Kaehler manifolds, and holomorphically pseudosymmetric Kaehler manifolds. Do they differ in some way from their Riemannian analogues? Yes, we prove they can all be characterized only in terms of holomorphic planes. Furthermore, the concept of holomorphically pseudosymmetric Kaehler manifold is different from the classical notion of pseudosymmetric Riemannian manifold proposed by Deszcz. We study some relations between both definitions of pseudosymmetry and the so called double sectional curvatures in the sense of Deszcz. We also present a geometric interpretation of the complex Tachibana tensor and a new characterization of constant holomorphic sectional curvature Kaehler manifolds.

In this paper, when N is a compact Riemannian manifold, we discuss the nonexistence of conformal deformations onRiemannian warped product manifold M =(a, ∞ )× f N with prescribed scalar curvature functions.Key words : Warped Product, Scalar Curvature, Partial Differential Equation 1. Introduction In a recent study [7-9] , M.C. Leung has studied theproblem of scalar curvature functions on Riemannianwarped product manifolds and obtained partial resultsabout the existence and nonexistence of Riemannianwarped metric with some prescribed scalar curvaturefunction. He has studied the uniqueness of positive solu-tion to equation(1.1)where is the Laplacian operator for an n-dimen-sional Riemannian manifold (N, g 0 ) and d n = n−2/4(n−1). Equation (1.1) is derived from the conformal defor-mation of Riemannian metric [1,4-6,8,9] .Similarly, let (N, g 0 ) be a compact Riemanniandimensional manifold. We consider the (n+1)−dimen-sional Riemannian warped product manifold M =(a, ∞)× f N with the metric g = dt

An isometric immersion of a Riemannian manifold M into a Riemannian manifold M is called helical if the image of each geodesic has constant curvatures which are independent of the choice of the particular geodesic. Suppose M is a compact Riemannian manifold which admits a minimal helical immersion of order 4 into the unit sphere. If the Weinstein integer of M equals that of one of the projective spaces, then M is isometric to that projective space with its canonical metric. 0. Introduction. In [2], Besse constructed a minimal immersion with a nice property of a strongly harmonic manifold into a sphere. This nice property is that the images of geodesics of the strongly harmonic manifold are of constant curvatures as curves in the sphere, and the curvatures and the osculating orders are independent of geodesics. Immersions with such a property are said to be helical (cf. [8]). Sakamoto [8] studied a helical immersion of a Riemannian manifold M into a unit sphere. The result is that if M is compact, then it is a Blaschke manifold and, moreover, if the helical immersion is minimal, then M is a globally harmonic manifold. It is well known that if a globally harmonic manifold is compact and simply connected, then it is a strongly harmonic manifold (Michel's theorem, cf. [2]). Thus, we can declare that the theory of helical minimal immersions of compact simply connected Riemannian manifolds into a unit sphere is a submanifold version of strongly harmonic manifold theory. Let f: M S(1) be a helical immersion of a compact Riemannian manifold M into a unit sphere S(1). Since M is a Blaschke manifold, all geodesics of M are simply closed and of the same length. Furthermore, if we denote the cut-locus of x E M by Cut(x), then the unit tangent vectors at x of geodesics emanating from x and entering to y E Cut(x) compose a great sphere in the unit tangent sphere at x. The dimension of the great sphere is independent of x and it is equal to 0, 1, 3, 7, or n 1 (n = dim M), which is the index of the first conjugate point y of x (cf. [2], Proposition 5.39 and Theorem 7.23). Geodesics are also helical in the Euclidean Received by the editors March 15, 1984 and, in revised form, June 20, 1984. 1980 Mathematics Subject Classification. Primary 53C40; Secondary 53C42.

We prove several Liouville-type non-existence theorems for higher order Codazzi tensors and classical Codazzi tensors on complete and compact Riemannian manifolds, in particular. These results will be obtained by using theorems of the connections between the geometry of a complete smooth manifold and the global behavior of its subharmonic functions. In conclusion, we show applications of this method for global geometry of a complete locally conformally flat Riemannian manifold with constant scalar curvature because its Ricci tensor is a Codazzi tensor and for global geometry of a complete hypersurface in a standard sphere because its second fundamental form is also a Codazzi tensor.

LetM be a compact Riemannian manifold with smooth boundary ∂M. We get bounds for the first eigenvalue of the Dirichlet eigenvalue problem onM in terms of bounds of the sectional curvature ofM and the normal curvatures of ∂M. We discuss the equality, which is attained precisely on certain model spaces defined by J. H. Eschenburg. We also get analog results for Kahler manifolds. We show how the same technique gives comparison theorems for the quotient volume(P)/volume(M),M being a compact Riemannian or Kahler manifold andP being a compact real hypersurface ofM.

Obata [3] gave a characterization up to isometry of the standard sphere sn in terms of the Hessian operator on a complete Riemannian manifold. With the convention Hess u = Vgrad u, his result says that if M is a complete Riemannian manifold which admits a nondegenerate function u such that Hess u = -u- Id then M is isometric to the standard sphere. Obata goes on to prove related results in conformal geometry which take advantage of the existence of a function whose Hessian has a special form. Other authors have also obtained strong geometric properties of a Riemannian manifold by exploiting the existence of a function u with Hess u = f - Id for some function f. In particular, this last equation implies that M is a warped product. For a proof and related results, see, for example, Osgood and Stowe [5]. In the complex case, a characterization of cn up to isometry was given by Stoll [6] via the complex Monge-Amp~re operator. Stoll's result says that if M is a complex manifold which admits a strictly plurisubharmonic exhaustion r: M --> [0, ~) such that (dd c log r) n -- 0, then (M, r) = (C", Iz12). In other words M, with hermitian metric given by the Kahler form ddCr, is biholomorphically isometric to C". Obata also showed that a complete, connected and simply connected Kahler manifold is isometric to the complex projective space ]?n if and only if it admits a solution to a certain linear system of third order differential equations [4]. Blair [1] subsequently showed that in some cases this characterization of ]?n follows from a corresponding result for Riemannian manifolds and indicated that one would not expect a characterization of ]?n by a Hessian equation analogous to that which Obata used to characterize S n. In this paper we give a complex analog of Obata's theorem [3]. We characterize complex projective space up to biholomorphic isometry by the existence of a solution to a system of second order equations. Since ]?n with the Fubini-Study metric is not a warped product, there does not exist a nontrivial function u on ]?n whose Hessian is a multiple of the identity. However, ]?n with one point deleted is the hyperplane section bundle over I? n- 1 and the fibers of this bundle are totally geodesic complex lines. Thus, there is a relationship between the natural metric structure and the line bundle structure of ~n. This relationship provides the motivation for the construction of a function u on ]?n

In this paper, we examine torse-forming vector fields to characterize extrinsic spheres (that is, totally umbilical hypersurfaces with nonzero constant mean curvatures) in Riemannian and Lorentzian manifolds. First, we analyze the properties of these vector fields on Riemannian manifolds. Next, we focus on Ricci solitons on Riemannian hypersurfaces induced by torse-forming vector fields of Riemannian or Lorentzian manifolds. Specifically, we show that such a hypersurface in the manifold with constant sectional curvature is either totally geodesic or an extrinsic sphere.