Showing papers on "Scalar curvature published in 1981"

Some regularity theorems in riemannian geometry

Abstract: © Gauthier-Villars (Éditions scientifiques et médicales Elsevier), 1981, tous droits réservés. L’accès aux archives de la revue « Annales scientifiques de l’É.N.S. » (http://www. elsevier.com/locate/ansens) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Curvature, diameter and Betti numbers.

Abstract: We give an upper bound for the Betti numbers of a compact Riemannian manifold in terms of its diameter and the lower bound of the sectional curvatures. This estimate in particular shows that most manifolds admit no metrics of non-negative sectional curvature.

On the upper estimate of the heat kernel of a complete riemannian manifold

Abstract: Let M be a complete non-compact Riemannian manifold whose sectional curvature is bounded between two constants -k and K. Then one expects that the heat diffusion in such a manifold behaves like the heat diffusion in Euclidean space. The purpose of this paper is to give a justification of such a statement. In [5], J. Cheeger and the third author found a lower estimate of the heat kernel by "comparing" it with the heat kernel of the space form whose curvature is the lower bound of the curvature of the manifold. This lower estimate is sharp if we insist the dependence should be on the lower bound of the Ricci curvature alone. It remains to give an upper estimate of the heat kernel. One does not, however, expect to have a comparison theorem for the upper bound because it is more sensitive to the geometry of the manifold. In fact, the heat kernel of the upper hialf space and the heat kernel of the complete hyperbolic manifold with finite volume have quite different behavior. This is reflected by the fact that the Laplacian has no discrete spectrum in the first case while infinite number of discrete eigenvalues may exist in the latter case. What we will prove here is that in any case, the heat kernel has to decay in a manner similar to the Euclidean heat kernel. Thus we will prove that for any constant C > 4, there exists C1 depending on C, T, the bound of the curvature of M and x so that for all t E [0, T]

An estimate for the curvature of bounded submanifolds

Abstract: 1. Statement of the result. All manifolds considered in this paper shall be connected, of class C"' (smooth), and dimension at least 2. Im- mersions will also be smooth, and of codimension at least 1. If M and M are Riemannian manifolds and (p: M - M is an isometric immersion with the property that sp(M) lies in a ball, we intend to estimate the supremum of the sectional curvature K of M in terms of the sectional curvature K of M and the radius of the ball. This is our result: THEOREM A. Let Ml' be a complete Riemannian manifold whose scalar curvature is bounded below; let M" +q be a Riemannian manifold with q c n -1, and Bx a closed normal ball in Mt4+q, of radius X. Sup- pose Sp:M" - j M"+q is an isometric immersion with the property that

The Riemannian geometry of the configuration space of gauge theories

Abstract: We state some new results about the configuration space of pure Yang-Mills theory. These results come from the study of the kinetic energy term of the Lagrangian of the theory. This term defines a riemannian metric on the space of non-equivalent gauge potentials. We develop a riemannian calculus on the configuration space, compute the riemannian connection, the curvature tensor, and solve for the geodesics, etc. We show that the Gribov ambiguity is more than an artefact of the choice of a gauge condition, and is related to the existence of conjugate points on the geodesics, and is thus an intrinsic feature of the theory.

The laplacian on asymptotically flat manifolds and the specification of scalar curvature

Abstract: It is shown that the Laplacian on an asymptotically flat manifold is an isomorphism between certain weighted Sobolev spaces. This is used to find a necessary and sufficient condition for an asymptotically flat metric to be conformally equivalent to one with vanishing scalar curvature. This in turn is used to give an example of a metric which cannot be conformally deformed within the class of asymptotically flat metrics to one with zero scalar curvature.

Path-integral evaluation of Feynman propagator in curved spacetime

Abstract: We develop an efficient approximation procedure for evaluating the scalar Feynman propagator in arbitrary spacetimes. In the familiar manner we represent it by an integral over the transition amplitude for a Schroedinger-type equation (proper-time method). The amplitude is then represented by a Feynman path integral which is dominated by the contribution of a certain extremal path. The contributions of adjacent paths are then simply expressed by working in Fermi normal coordinates based on the extremal path. In this manner the path integral becomes an ordinary multiple integral over ''Fourier coefficients'' which represent the various paths. For a conformal field, or for spacetimes with constant scalar curvature, we evaluate the integral in the Gaussian approximation in terms of the curvature along the (geodesic) extremal path. We show the result to be related to the Schwinger-DeWitt expansion for the amplitude, but valid for well-separated end points. In the Einstein universe our expression gives the exact amplitude and propagator. In the de Sitter spacetime it gives a good approximation for the amplitude even for well-separated points. We also evaluate the post-Gaussian corrections to the amplitude, though we do not implement them in a concrete spacetime. For nonconformal fields in spacetimes with varying scalarmore » curvature, we evaluate the amplitude in the Gaussian approximation in terms of the values of the curvature along the extremal (nongeodesic) path. It is very different in form from the one mentioned earlier, which suggests the existence of novel effects arising from variation in the scalar curvature.« less

An inequality between the exterior diameter and the mean curvature of bounded immersions

Abstract: This paper is a natural outgrowth of [9], where the problem of studying the boundedness properties of complete minimal surfaces in 1R" was approached via the consideration of a certain gradient flow. After the completion of [9] it was soon realized that the basic technique (Lemma 3 below) could be successfully employed to study boundedness of arbitrary complete submanifolds. In this regard, the method can be used to prove the following three theorems.

Quantum unified field theory from enlarged coordinate transformation group. II

Abstract: If the relativity principle, which states that the law of propagation for light has the same form for all macroscopic observers, is extended to include quantum observers, this leads directly to the quantum unified field theory which was introduced in a previous paper. This theory appears suitable for describing all known interactions. Gravitation and electromagnetism are described by the Einstein equations ${G}_{\ensuremath{\mu}\ensuremath{ u}}=\frac{1}{2}({e}_{\ensuremath{\mu}\ensuremath{ u}}\ensuremath{-}{K}_{\ensuremath{\mu}}{j}_{\ensuremath{ u}}\ensuremath{-}{K}_{\ensuremath{ u}}{j}_{\ensuremath{\mu}})\ensuremath{-}R{K}_{\ensuremath{\mu}}{K}_{\ensuremath{ u}}$, where ${G}_{\ensuremath{\mu}\ensuremath{ u}}$ is the Einstein tensor, $R$ is the Ricci scalar, ${e}_{\ensuremath{\mu}\ensuremath{ u}}$ is the usual stress-energy tensor for the free electromagnetic field, and ${j}_{\ensuremath{\mu}}$ is the electromagnetic current. The vector ${K}_{\ensuremath{\mu}}$ plays a dual role. It is the electromagnetic vector potential in the covariant Lorentz gauge, and, it is also a unit timelike vector interpretable as the velocity of the observer.

On Stokes’ theorem for noncompact manifolds

Abstract: Stokes' theorem was first extended to noncompact manifolds by Gaffney. This paper presents a version of this theorem that includes Gaffney's result (and neither covers nor is covered by Yau's extension of Gaffney's theorem). Some applications of the main result to the study of subharmonic functions on noncompact manifolds are also given. 0. In [4] Gaffney extended Stokes' theorem to complete Riemannian manifolds Mn and proved that if w E A` l(M') and dw are both integrable then fIm dc = 0. The same conclusion was established by Yau under the sole condition that lim inflror rf B(rIZw d vol = 0, where B(r) = geodesic ball of radius r about some pointp E M (cf. the remarks in the appendix to [13]). The purpose of this note is to give another extension of Gaffney's theorem that covers some cases not included in Yau's result. 1. Before formulating our main theorem we recall some background material. Let Mn be a Riemannian n-manifold and suppose that, in local coordinates (xl, ... , x'), the vector field X is represented as X = Xi a/ax', the metric tensor is given as (ga), and g = det(ga). The quantities given locally as Vg and :(a1ax')(V_g_X') are then densities (i.e., under a coordinate change they are multiplied by the absolute value of the Jacobian corresponding to the change of coordinates), and hence they may be unambiguously integrated via local coordinates even if Mn is not orientable ([10], cf. pp. 21-26 of [9] where a different terminology is used). When f is a scalar (i.e., has a local expression that is invariant under change of coordinates), we will denote the integral of the density fVg by f f dvg. If M' is an oriented Riemannian manifold, then Vjdxl A*** Adx' represents the volume form vg in local coordinates, and for the scalar (l1/Vg)(a/axi)(V-gX'), called the divergence of X and denoted divg X, we have J m divg Xvg = IM divg Xdvg. Here the integrand on the left is an n-form, and the integrand on the right is actually the density Vj* divg X. On any oriented manifold with volume form X > 0, one defines the divergence of X with respect to w, denoted div, X, via d(ixo) = div, Xw, where ix denotes contraction with X (cf. [8]), and it is easily seen that divg X = div. X if X = vg for Received by the editors September 5, 1980. 1980 Mathematics Subject Classification. Primary 58G99, 53C99; Secondary 53C20, 58C99.

On a geometric quantization scheme generalizing those of Kostant-Souriau and Czyz

Abstract: A quantization method (strictly generalizing the Kostant-Souriau theory) is defined, which may be applied in some cases where both Kostant-Souriau prequantum bundles and metaplectic structures do not exist. It coincides with the Czyz theory for compact Kahler manifolds with locally constant scalar curvature. Quantization of dynamical variables is defined without use of intertwining operators, extending either the Kostant map or some ordering rule like that of Weyl or Born-Jordan.

Weyl-Dirac theory and superconductivity

Abstract: The field equations of the Weyl-Dirac theory with complex Dirac scalar suggest a Lagrangian constraint which leads to electromagnetic-flux quantization As a consequence, the Weyl geometry is that of a charged, multiply connected superconductor With the retention of the cosmological constant λ there arises a broken symmetry of Higgs type This causes the photon to acquire a mass varying as\(( - 2\lambda /\bar R)^{\tfrac{1}{2}} \), where\(\bar R\) is the scalar curvature

On the point spectrum for finite volume symmentric spaces of negative curvature

Abstract: (1981). On the point spectrum for finite volume symmentric spaces of negative curvature. Communications in Partial Differential Equations: Vol. 6, No. 9, pp. 963-992.

La courbure scalaire des varietes riemanniennes

Abstract: © Association des collaborateurs de Nicolas Bourbaki, 1981, tous droits réservés. L’accès aux archives du séminaire Bourbaki (http://www.bourbaki. ens.fr/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.

Compact riemannian manifolds with harmonic curvature and non-parallel ricci tensor

Abstract: While every manifold with parallel Ricci tensor has harmonic curvature (i.e. satisfies 6R = 0), there are examples ([3], Theorem 5.2) of open Riemannian manifolds with 6R = 0 and VS ~ O. In [1] J.P. Bourguignon has asked the question whether the Ricci tensor of a compact Riemannian manifold with harmonic curvature must be parallel. The aim of this note is to describe an easy example answering this question in the negative. More precisely, metrics with 6R = 0 and VS ~ 0 are exhibited on S 1 × N 3, N 3 being e.g. the 3-sphere or a lens space. By taking products of these manifolds with themselves or with arbitrary compact Einstein manifolds, one gets similar examples in all dimensions greater {han three.

Total Scalar Curvature of Tubes about Curves

Abstract: Let A!l be a RIEMANNian manifold of class c\" and G,(s) a geodesic sphere with center m c M and radius s. In [8] A. GRAY and the second author determined a power series expansion for the volume S,(s) of G,(s). The main purpose of [8] is t o try to characterize EucLIDean space and the rank one symmetric spaces by means of the volume function S,(s). For example the authors consider the following conjecture : Let M be an n-dhensional RIEMANNian manifold of class c\" and suppose that for all mc &I and all sufficiently small geodesic spheres the volume S,(s) is the same as for EucLIDean space, i.e. X,(s) =c,-,slz-'. Then M i s locally flat. (Here cnP1 denotes the volume of the ( T L 1)-dimensional unit sphere in P.) Similar conjectures are given for the rank one symmetric spaces. These questions are answered affirmatively in many important cases but the general problem remains open. In addition B.-Y. CHEN and the second author treated the differential geometry of geodesic spheres extensively in [4] and [ 5 ] where a lot of similar problems are studied. In these papers the authors determine mainly properties of G,(s) considered as a submanifold of the ambient space iM. Next let CT be a topologically embedded curve of finite length L(a) in JI and denote by P, a tube ahout (T with sufficiently small radius s to avoid focal points of (T. In a subsequent paper [9] a power series expansion is given for the volume S,(s) of P, and similar conjectures are studied; for example: Let Llf be an %dimensional RIEMANNian manifold of class c\" and suppose that for all sufficiently short geodesics 0 the volume S,(s) of all sufficiently small tubes P, about (T is the same as for Eucunean space, i.e. S,(s)=c,-,sn-'L(a). Then &! is

On the conservation laws in the theory of fields in Finsler spaces

Abstract: Some conservation laws are obtained for the fields in some special Finsler spaces such as scalar curvature space, locally Minkowski space, etc.

Necessary and sufficient conditions for existence of minimal hypersurfaces in spaces of constant curvature

Abstract: 1.1. Let (M2, ds 2) be a two-dimensional Riemannian manifold M 2 with metric ds 2 and let p ~ M be a point where the Gaussian curvature K of ds 2 is nonpositive. The Ricci theorem [2, pg. 41 l] states that a necessary and sufficient condition for some neighborhood of p in M to be isometrically and minimally immersed in the euclidean space R 3 is that the metric ~, K ds 2 be flat around p. This condition, usually called the Ricci condition, can be shown to be equivalent to the fact that the metric (- K) ds 2 has constant Gaussian curvature equal to one. The purpose of the present paper is to obtain a generalization of the latter version of the Ricci condition for hypersurfaces of a space M"+l(c) of constant curvature c, c any real number. More precisely, we prove Theor. (1.2) below, for the statement of which we need some notation. We will denote by RicM the quadratic form in a Riemannian manifold (M, ds 2) defined by the Ricci tensor of ds 2. (,) and V will denote the inner product and the Levi-Civitta connection, respectively, of ds 2. We will be considering the case where (n-~ 1)(-RicM+cds 2) is positive definite: then (n- I)(-RicM + cds 2 ) defines a further Riemannian metric on M, and we will denote by ((,)) and ~' the inner product and the Riemannian connect'ion, respectively, of such a metric. Finally, S(M) will denote the bundle over M whose fiber at q ~ M in the space of symmetric linear maps of T~(M).