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Showing papers on "Scalar curvature published in 1980"


Journal ArticleDOI
TL;DR: In this paper, it was shown that stable minimal surfaces in Riemannian 3-manifolds can be expressed analytically by the condition that o n any compact domain of M, the first eigenvalue of the operator A+Ric(v)+AI* be positive.
Abstract: The purpose of this paper is to study minimal surfaces in three-dimensional manifolds which, on each compact set, minimize area up to second order. If M is a minimal surface in a Riemannian three-manifold N, then the condition that M be stable is expressed analytically by the requirement that o n any compact domain of M, the first eigenvalue of the operator A+Ric(v)+(AI* be positive. Here Ric (v) is the Ricci curvature of N in the normal direction to M and (A)’ is the square of the length of the second fundamental form of M. In the case that N is the flat R3, we prove that any complcte stable minimal surface M is a plane (Corollary 4). The earliest result of this type was due to S. Bernstein [2] who proved this in the case that M is the graph of a function (stability is automatic in this case). The Bernstein theorem was generalized by R. Osserman [lo] who showed that the statement is true provided the image of the Gauss map of M omits an open set on the sphere. The relationship of the stable regions on M to the area of their Gaussian image has been studied by Barbosa and do Carmo [l] (cf. Remark 5 ) . The methods of Schoen-Simon-Yau [ 113 give a proof of this result provided the area growth of a geodesic ball of radius r in M is not larger than r6. An interesting feature of our theorem is that it does not assume that M is of finite type topologically, or that the area growth of M is suitably small. The theorem for R3 is a special case of a classification theorem which we prove for stable surfaces in three-dimensional manifolds N having scalar curvature SZO. We use an observation of Schoen-Yau [8] to rearrange the stability operator so that S comes into play (see formula (12)). Using this, and the study of certain differential operators on the disc (Theorem 2), we are

727 citations



Journal ArticleDOI
TL;DR: In this paper, it is shown on examples that the distance between nearby states is related to quantum fluctuations; in particular, in the particular case of the harmonic oscillator group the condition of zero curvature appears to be identical to that of non dispersion of wave packets.
Abstract: A metric tensor is defined from the underlying Hilbert space structure for any submanifold of quantum states. The case where the manifold is generated by the action of a Lie group on a fixed state vector (generalized coherent states manifold hereafter noted G.C.S.M.) is studied in details; the geometrical properties of some wellknown G.C.S.M. are reviewed and an explicit expression for the scalar Riemannian curvature is given in the general case. The physical meaning of such Riemannian structures (which have been recently introduced to describe collective manifolds in nuclear physics) is discussed. It is shown on examples that the distance between nearby states is related to quantum fluctuations; in the particular case of the harmonic oscillator group the condition of zero curvature appears to be identical to that of non dispersion of wave packets.

520 citations



Journal ArticleDOI
TL;DR: In this article, the authors apply adiabatic regularisation to a scalar field propagating in a Robertson-Walker universe with arbitrary coupling to the scalar curvature.
Abstract: Adiabatic regularisation is applied to a scalar field propagating in a Robertson-Walker universe with arbitrary coupling to the scalar curvature. Explicit expressions for the expectation value of the quantum stress tensor in an adiabatic vacuum are obtained. This calculation yields the terms which are to be subtracted from the divergent mode-sum expressions for expectation values of the stress tensor to give a finite, renormalised stress tensor. It is shown that the removal of the infinite terms in this subtraction procedure corresponds to the renormalisation of coupling constants in Einstein's equation. A short description is given of the way in which adiabatic regularisation produces a trace anomaly.

177 citations




Journal ArticleDOI
TL;DR: In this article, the Ricci tensor of a compact Riemannian manifold with harmonic curvature must be parallel, and the authors give examples of RiemANNIAN manifolds with fiR = O, VS + 0, and such that the RICCI tensor S has at any point less than three distinct eigenvalues.
Abstract: S being the Ricci tensor. While every manifold with parallel Ricci tensor has harmonic curvature, i.e., satisfies fiR=O, there are examples ([3], Theorem 5.2) of open Riemannian manifolds with fiR=O and VS+O. In [1] Bourguignon has asked the question whether the Ricci tensor of a compact Riemannian manifold with harmonic curvature must be parallel. The aim of this paper is to give examples (see Remark 2) answering this question in the negative. All our examples are conformally flat (Corollary 1). Moreover, we obtain some classification results, restricting our consideration to Riemannian manifolds with fiR = O, VS + 0 and such that the Ricci tensor S has at any point less than three distinct eigenvalues. Starting from a description of their local structure at generic points (Theorem 1), we find all four-dimensional, analytic, complete and simply connected manifolds of this type (Theorem 2). They are all non-compact, but some of them do possess compact quotients. Next we prove (Theorem 3) that all compact four-dimensional analytic Riemannian manifolds with the above properties are covered by S 1 x S 3 with a metric of an explicitly described form. Throughout this paper, by a manifold we mean a connected paracompact manifold of class C ~ or analytic. By abuse of notation, concerning Riemannian manifolds we often write M instead of (M,g) and @ , v ) instead of g(u,v) for tangent vectors u, v.

73 citations


Journal ArticleDOI
TL;DR: In this article, a generalisation of normal ordering to curved space-time is introduced, based on the construction of adiabatic particle states in Robertson-Walker space time.
Abstract: Renormalisation of lambda phi 4 theory in curved space-time is considered in the interaction picture. A generalisation of normal ordering to curved space-time is introduced, based on the construction of adiabatic particle states in Robertson-Walker space-time. Dimensional regularisation is used to define uniquely the divergent quantities which are removed by normal ordering. It is shown that this normal ordering is sufficient to make finite all physical processes including vacuum polarisation to first order in lambda . An alternative and equivalent procedure is given which requires renormalisation of the mass and of the constant which couples the field to the Ricci scalar. The stress tensor is found to be finite to first order in lambda and it is shown that if the free-field theory in a Robertson-Walker universe predicts that particles are created by the gravitational field with a black-body spectrum then this spectrum is maintained when first-order self-interactions are taken into account. Finally, some aspects of the renormalisation of second-order physical processes are discussed.

70 citations


Journal ArticleDOI
TL;DR: In this paper, the density properties of subgroups D~_I(H) that satisfy the duality condition (defined below) were considered. But the density results are very similar to those of [5].
Abstract: Let H denote a complete simply connected Riemannian manifold of nonpositive sectional curvature, and let I(H) denote the group of isometries of H. In this paper we consider density properties of subgroups D~_I(H) that satisfy the duality condition (defined below), These density properties also yield characterizations of Riemannian symmetric spaces of noncompact type and results about lattices in H that strengthen several of the results of [ 11 ] and [ 15]. If H is a symmetric space of noncompact type and if D is a subgroup of Io(H), then the duality condition for D is implied by the Selberg property (S) for D [20, pp. 4-6] or [10]. A partial converse is obtained in [10]. It is an interesting question whether the two conditions are equivalent in this context. Our density results are very similar to those of [5]. In Proposition 4.2 we obtain a differential geometric version of the Borel density theorem (cf. Corollary 4.2 of [5]): Let H admit no Euclidean de Rham factor, and let G~_I(H) be a subgroup whose normalizer D in I(H) satisfies the duality condition. Then either (1) G is discrete or (2) there exist manifolds Hi , / /2 such that (a) H is isometric to the Riemannian product HlXH2, (b) H1 is a symmetric space of noncompact type, (c) ((~)0=Io(Hl) and (d) there exists a discrete subgroup B~_I(Hz), whose normalizer in 1(//2) satisfies the duality condition, such that Io(HO• is a subgroup of t) of finite index in 0 . Using the result just quoted or the main theorem of section 3 we then obtain the following decomposition of a manifold H whose isometry group I(H) satisfies the duality condition (Proposition 4.1): Let I(H) satisfy the duality condition. Then there exist manifolds H0, Ht and H2, two of which may have dimension zero, such that (1) H is isometric to

62 citations



Journal ArticleDOI
TL;DR: In this article, the integral of the Ricci curvature of the tangent vector field of a complete geodesic in a Riemannian manifold or a complete nonspacelike geodesics in a Lorentzian manifold is shown to contain a pair of conjugate points.
Abstract: Generalizing results of Cohn-Vossen and Gromoll, Meyer for Riemannian manifolds and Hawking and Penrose for Lorentzian manifolds, we use Morse index theory techniques to show that if the integral of the Ricci curvature of the tangent vector field of a complete geodesic in a Riemannian manifold or of a complete nonspacelike geodesic in a Lorentzian manifold is positive, then the geodesic contains a pair of conjugate points. Applications are given to geodesic incompleteness theorems for Lorentzian manifolds, the end structure of complete noncompact Riemannian manifolds, and the geodesic flow of compact Riemannian manifolds.


Journal ArticleDOI
TL;DR: In this paper, the authors considered the problem of time-independent surfaces of prescribed mean curvature, and showed that no physical problem has been known involving time-dependent surfaces, except the still unsolved and even nonattacked corresponding hyperbolic problem.

Journal ArticleDOI
TL;DR: In this article, it was shown that the spectrum of a riemannian manifold gives a lot of information about its structure, but it alone does not completely determine the structure in general.
Abstract: Introduction. Let M be a compact connected riemannian manifold and Δ the Laplacian acting on the space of C°°-funcΐions on M. The operator Δ has a discrete spectrum consisting nonnegative eigenvalues with finite multiplicities. We denote by Spec M the spectrum of the Δ. Two compact connected riemannian manifolds M and N are said to be isospectral to each other if Spec M=SpeciV. The spectrum of a riemannian manifold gives a lot of information about its riemannian structure, but it does not completely determine the riemannian structure in general. In fact, there exist two flat tori which are isospectral but not isometric (by J. Milnor, see [1]). On the other hand some distinguished riemannian manifolds are completely characterized by their spectra as riemannian manifolds. The n-dimensional sphere S with the canonical metric and the real projective space P(R) with the canonical metric are completely characterized by their spectra as riemannian manifolds if wfgό (see [1], [8]). Recently, it has been shown successively that a 3-dimensional lens space M is completely determined by its spectrum as a riemannian manifold; first by M. Tanaka [7] in the case the order ITΓ^M)! of the fundamental group of M is odd prime or 2-times odd prime, then by the author and Y. Yamamoto [4] in a more general case, and finally by Y. Yamamoto [11] without any restriction. These examples are riemannian manifolds of positive constant curvature. A connected complete riemannian manifold M ( d i m M ^ 2 ) of positive constant curvature 1 is called a spherical space form. Now, we consider the problem;

Journal ArticleDOI
TL;DR: In this paper, it was shown that every analytic surface in a euclidean space with parallel normalized mean curvature vector must either lie in aE4 or lie in hypersphere ofEm as a minimal surface.
Abstract: A surfaceM in a Riemannian manifold is said to have parallel normalized mean curvature vector if the mean curvature vector is nonzero and the unit vector in the direction of the mean curvature vector is parallel in the normal bundle. In this paper, it is proved that every analytic surface in a euclideanm-spaceEm with parallel normalized mean curvature vector must either lies in aE4 or lies in a hypersphere ofEm as a minimal surface. Moreover, it is proved that if a Riemann sphere inEm has parallel normalized mean curvature vector, then it lies either in aE3 or in a hypersphere ofEm as a minimal surfaces. Applications to the classification of surfaces with constant Gauss curvature and with parallel normalized mean curvature vector are also given.


Journal ArticleDOI
TL;DR: In this paper, a computer-aided classification of geometries in terms of the curvature tensor and its covariant derivatives is presented, which extends the Petrov classification to a complete classification of the local geometry and the dimensions of the group of motions and its isotropy subgroup are obtained.

Journal ArticleDOI
TL;DR: In this paper, the curvature identities and holomorphic sectional curvatures of locally conformal Kihler manifolds are investigated and sufficient conditions for such manifolds to be globally conformal kiihler are derived.
Abstract: Curvature identities and holomorphic sectional curvature of locally conformal Kihler manifolds are investigated. Particularly, sufficient conditions for such manifolds to be globally conformal Kiihler are derived. In [2], A. Gray showed the importance of various curvature identities in the geometry of the almost Hermitian manifolds. It is therefore to be expected that such identities should play some role in the geometry of the locally conformal Kahler (l.c.K.) manifolds [4], [5], too. Particularly, we can ask whether such identities could characterize the Kahler manifolds in the class of the l.c.K. manifolds and we want to give here a theorem of this kind. We shall also obtain information about the Kdhler-nullity distribution [2] and about the holomorphic sectional curvature of the l.c.K. manifolds. 1. L.c.K. manifolds. A. Gray studied in [2] various curvature identities which could appear in the almost-Hermitian geometry. As usual, we are working in the COO-category, and we shall denote by M2n (n > 1) the manifold, by J its complex structure, by g the Hermitian metric and by Q(X, Y) = g(X, JY) (X, Y, . . . are always vector fields on M) its fundamental form. We refer, for instance, to [3] for a detailed exposition of the almost Hermitian geometry. Now, if e& is some class of almost-Hermitian manifolds, Gray defines the classes fi (i = 1, 2, 3) as its subclasses characterized respectively by the identities [2]: (1) RXYZW = Rxy(Jz)(JW), (2) RXYZW R(JX)(JY)ZW R(JX)Y(JZ)W R(Jx)yz(JW) = 0, (3) RXYZW = R(JX)(JYxJZxJW), where R is the curvature tensor of the Levi-Civita connection V of g. It is known that for every e one has El C E2 C f C e and that the class YC of the Kahler manifolds satisfies all of the relations (1), (2), (3) [2]. Particularly, if we denote by C, Yc the class of the l.c.K. manifolds, we can speak of the classes CuCi (i = 1, 2, 3) and we shall investigate some relations between these classes. But, before doing this, we recall some properties of the manifolds of C Yc and we establish some useful results including the determination of their Kahler-nullity distribution [2]. More about this subject can be found in [5]. Received by the editors January 8, 1979 and, in revised form, June 10, 1979. 1980 Mathematics Subject Classification. Primary 53C55. ? 1980 American Mathematical Society 0002-9947/80/0000-0255 /$0 3.25



Journal ArticleDOI
TL;DR: In this article, it was shown that a compact K/ihler manifold of positive holomorphic bisectional curvature must be biholomorphic to the complex projective space.
Abstract: After proving the dimension two case jointly with Andreotti, Frankel [3] conjectured that a compact K/ihler manifold of positive sectional curvature is biholomorphic to the complex projective space. Mabuchi [8] verified the case of dimension three by using the result of Kobayashi-Ochiai [6]. Very recently by using the methods of algebraic geometry of positive characteristic Mori [10] proved that a compact Kihler manifold with ample tangent bundle must be biholomorphic to the complex projective space. By methods of Kihler geometry Siu-Yau [12] proved that a compact K/ihler manifold of positive holomorphic bisectional curvature must be biholomorphic to the complex projective space. Frankel’s conjecture is a special case of these more general results. It is reasonable to conjecture that there are similar curvature characterizations for other irreducible compact symmetric K/ihler manifolds. In this paper we obtain such a curvature characterization for the complex hyperquadric. Definition. Let M be a K/ihler manifold and P M. If the holomorphic bisectional curvature of M is nonnegative at P, then for a nonzero element (a) of the holomorphic tangent space T,M of M at P, the curvature null space at P in the direction of , denoted by N,(O, is defined as the set of all

Journal ArticleDOI
01 Feb 1980
TL;DR: In this article, the results of M. Tani and M. Okumura on compact hypersurfaces of Eucidean space are extended to complete spaces by an application of S.-T. Yau's "maximum principle".
Abstract: Results of M. Tani on compact conformally flat manifolds and of M. Okumura on compact hypersurfaces of Eucidean space are extended to complete spaces by an application of S.-T. Yau's "maximum principle".



Journal ArticleDOI
TL;DR: In this paper, the curvature tensor and a finite number of its covariant derivatives relative to a field of orthogonal frames are discussed and compared to the metrical tensor.
Abstract: A coordinate-invariant description of a Riemannian manifold is known to be furnished by the curvature tensor and a finite number of its covariant derivatives relative to a field of orthogonal frames. These tensors are closer to measurements than the metrical tensor is. The present article discusses this description's usefulness in general relativity and the redundancy among the curvature tensor and its derivatives.


Journal ArticleDOI
TL;DR: In this article, a systematic study of spherically symmetric space-time manifolds with respect to the eigenvalues of the Ricci tensor is made, and the cases of two double or one quadruple eigenvalue are treated exhaustively.
Abstract: In view of the geometrical importance of spaces with constant scalar curvature, a systematic study of spherically symmetric such space–time manifolds with respect to the eigenvalues of the Ricci tensor is made. The cases of two double or one quadruple eigenvalue are treated exhaustively. In the generic case of one double and two single eigenvalues, no conformally flat solutions, and only solutions with one arbitrary function of one variable are found. We also give all four-dimensional decomposable s.s. spaces with constant curvature scalar.

Journal ArticleDOI
01 Jan 1980
TL;DR: In this article, it was shown that a compact n-dimensional Riemannian manifold whose sectional curva-tures are everywhere less than constant X-2 cannot be isometrically immersed into euclidean space of dimension 2n -1 so as to be contained in a ball of radius X. The special case where M is compact is due to Jacobowitz.
Abstract: Let AY be a complete Riemannian manifold of dimension n, with scalar curvature bounded from below. If the isometric immersion of M into euclidean space of dimension n + q, q X-2. The special case where M is compact is due to Jacobowitz. Generalizing results by Tompkins, Chern and Kuiper, and Otsuki, Jacobowitz proved that a compact n-dimensional Riemannian manifold whose sectional curva- tures are everywhere less than constant X-2 cannot be isometrically immersed into euclidean space of dimension 2n - 1 so as to be contained in a ball of radius X (see (1) and the references therein). In this note we shall prove a quantitative result concerning isometric immersions, which includes Jacobowitz's theorem as a special case. The proof of our result will consist in a simple application of a theorem by