DOCSLIB.ORG

Isometric Embeddings Between Space Forms

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Isometric Embeddings Between Space Forms Isometric Embeddings between Space Forms David Brander A THESIS in Mathematics Presented to the Faculties of the University of Pennsylvania in Partial Ful¯llment of the Requirements for the Degree of Master of Philosophy 2003 Supervisor of Thesis Graduate Group Chairperson Acknowledgements Special thanks are due (in no particular order) to Professor Wolfgang Ziller for taking on the task of dissertation advisor and giving me advice whenever I needed it, to Dr. Ben Andrews for many helpful discussions, to Professor Chris Croke for helping me out with the ¯nal details of the dissertation, to Professor Dennis DeTurck for acting on my examination committee, to Professor David Harbater for always being willing to provide helpful advice during the time I spent at Penn, to all the administrative sta® in the Mathematics Department at Penn, who are always willing to help in any way they can, to all my friends among the graduate students at Penn (who know who they are) and especially to Cheni Wang for her love and support. ii ABSTRACT Isometric Embeddings between Space Forms David Brander Advisor: Wolfgang Ziller In 1901 D. Hilbert proved that there is no global isometric immersion of the hyperbolic plane into 3-dimensional Euclidean space, despite the fact that there is a local isometric embedding. Today it is known that there is a global isometric immersion of H2 into E5 and a global isometric embedding into E6, but it is an open problem whether these codimensions can be reduced. More generally, one can ask what the minimal codimensions are for a local embedding, complete immersion and n n+k global embedding of a simply connected space form Qc into another Qc~ , where the curvatures c and c~ are not equal. This thesis seeks to present the best results currently available with regard to this question. In particular, we will show that the problem is solved for all cases apart from global immersions and embeddings of negative extrinsic curvature (c < c~), when c = 0. For these unresolved cases we show 6 that the upper bounds for the minimal codimensions are quite small by constructing n explicit embeddings and immersions. The worst case is that of embedding H 1 into ¡ n+k 2 H c2 , for 1 < c < 0, where we only know that n 1 k 5n 5, and the best ¡ ¡ ¡ ¡ · · ¡ n n+k unresolved case is that of immersing a sphere SR into a smaller sphere Sr , where we know that the minimal codimension is either n or n + 1. We also present those non-existence results which are currently known. iii Contents 1 Introduction 1 1.1 Notation and Preliminaries . 3 2 Positive Extrinsic Curvature - Umbilic Hypersurfaces 5 2.1 0 < ~c < c: Spheres inside Larger Spheres . 6 2.2 ~c = 0 < c: Spheres into Euclidean Space . 6 2.3 ~c < 0; ~c < c : Hypersurfaces of Hn . 6 3 Negative Extrinsic Curvature: Local Results 10 3.1 A Non-Existence Result . 10 3.2 The Second Fundamental Form . 13 3.3 Explicit Local Embeddings . 20 3.3.1 0 < c < ~c: A Portion of a Sphere Embedded into a Smaller Sphere . 21 3.3.2 c = 0 < ~c: The Cli®ord Torus . 23 3.3.3 c < 0: Local Embeddings of Hyperbolic Space . 23 iv 4 Complete Immersions of Negative Extrinsic Curvature 26 4.1 0 = c < ~c: Euclidean into Spherical . 26 4.2 0 < c < ~c: Spheres into Spheres . 27 4.3 c < 0 : Immersing Hyperbolic Space - Some Constructions . 27 4.4 Non-Immersibility of the Hyperbolic Plane . 29 5 Global Embeddings of Negative Extrinsic Curvature 33 5.1 0 < c < ~c: Spheres inside Smaller Spheres . 33 5.2 c 0 < ~c : Hn and En into a Sphere . 36 · n 6n 6 5.3 Embedding H into E ¡ . 37 n 6n 5 5.4 Embedding H into H~c ¡ . 40 6 Conclusion 41 6.1 Open Problems . 42 A Smoothed Step Functions 44 v Chapter 1 Introduction A space form is a complete connected Riemannian manifold of dimension n 2 and ¸ constant sectional curvature c. Up to isometry, there is a unique simply connected n space form of dimension n and sectional curvature c, denoted here as Qc . The special cases where c is -1, 0 or 1 are the standard hyperbolic, Euclidean and spherical spaces, denoted Hn, En and Sn respectively. In this thesis we will address the question: when is it possible to isometrically embed, immerse or locally embed Qn into Qn+k, where c = c~. In particular, we would c c~ 6 like to know for each case what is the smallest codimension k for which this can be done. Naturally, this codimension must be at least one, because the two manifolds have di®erent sectional curvature. The class of di®erentiability we are considering is always C1. One can obtain isometric embeddings in very low codimensions of class C1 [14], but they do not preserve the di®erential geometric nature of the manifold 1 in question, because curvature comes from the second derivative of the metric. It is worthy of note that some of the global results we will give (speci¯cally the immersions and embeddings of the hyperbolic space with negative intrinsic curvature) are not real analytic. n n+k If Mc is a space form immersed into another Mc~ , then the di®erence in sectional curvatures, c c~, will be called the extrinsic curvature. It turns out that the problem ¡ of embeddings with positive extrinsic curvature is quite easy, and there is always a global embedding in codimension one. Therefore, the majority of this thesis is a discussion of the more problematic case of negative extrinsic curvature, where, in most cases, only the local problem has really been given a sharp answer. An example of the kind of thing that can happen is the following: the map f : ( ; 0) R E 3 ¡1 £ ! given by x f(x; y) = [ p1 e2tdt; exeiy]; Z0 ¡ where we have identi¯ed E3 with R C, is an immersion with constant sectional £ curvature 1. It looks like a horn enclosing the x axis in E3, with the wide end a ¡ 1 circle in the x -x plane and the pointed end at x = . It is a part of the hyperbolic 2 3 1 ¡1 plane, but not the whole, because it is not complete, being clearly incomplete as a metric subspace of E3. It is a famous theorem of Hilbert that there is in fact no complete immersion of H2 into E3 - even though there is a local embedding. It is also an interesting (and apparently still open) question whether the lowest codimension needed for a complete immersion is the same as that required for a global embedding. 2 It has been known for some time that there is a C 1 isometric embedding of any Riemannian manifold into En for su±ciently large n [15], [8]. Since, as will be shown below, En can be isometrically embedded into H n+1, and also realised as a one-to-one isometric immersion into a sphere of any radius of dimension 4n 1, this means that ¡ n+k any Riemannian manifold can be isometrically embedded into Qc , for any c, if k is large enough, with the caveat that if the manifold is not compact, and if c > 0, then \embedding" must be replaced with \one-to-one immersion". However, these general embedding theorems require a very large codimension, and it turns out that, for speci¯c cases, much better results are obtained by constructing explicit embeddings. Finally, why is it interesting to isometrically embed only simply connected space forms? The answer is that this is a natural ¯rst step to the general problem of n n embedding space forms, because any space form Mc has Qc as its universal cover. n n+k Thus, for example, an isometric embedding of Qc into Qc~ automatically gives an n n+k isometric immersion of Qc into any space form Mc~ . On the other hand, in order to n n+k have a chance of embedding Mc into Qc~ , it is necessary that there be an immersion n of Qc , which would be the lift of the desired embedding. 1.1 Notation and Preliminaries n Mc always stands for a Riemannian manifold of dimension n and constant sectional curvature c. In addition to the designations mentioned above, it is also common to n n use, for the sphere of radius r, the notation Sr instead of Q 1 , and when c < 0, r2 3 n n Hc = Qc . A metric is usually denoted by g and the second fundamental form by h. When a manifold is immersed in another manifold with the induced metric we will usually use g for the metric on both spaces. It is often convenient to use the notation ei® for the vector [cos ®; sin ®]. We will generally assume those de¯nitions and results commonly taught in ¯rst year graduate courses on Geometry and Topology. Assumptions more speci¯c to Riemannian Geometry can be found, for example, in [7]. The most important of these are the de¯nition of the curvature tensor R, which to every X and Y in the tangent space TpM assigns a mapping R(X; Y ) : TpM TpM given by ! R(X; Y )Z = X Y Z Y X Z Z; r r ¡ r r ¡ r[X;Y ] (where the expression does not depend on the extension of X, Y and Z to vector ¯elds in a neighbourhood of p), and Gauss' equation g(R(X; Y )Z; T ) = g(R~(X; Y )Z; T ) + g(h(X; T ); h(Y; Z)) g(h(Y; T ); h(X; Z)); ¡ which relates, through the second fundamental form h, the curvature tensor R~ of a manifold M~ to the curvature tensor R of a submanifold M.
Load more

Recommended publications
  • Compact Clifford-Klein Forms of Symmetric Spaces
    Compact Clifford-Klein forms of symmetric spaces – revisited In memory of Professor Armand Borel Toshiyuki Kobayashi and Taro Yoshino Abstract This article discusses the existence problem of a compact quotient of a symmetric space by a properly discontinuous group with emphasis on the non-Riemannian case. Discontinuous groups are not always abun- dant in a homogeneous space G/H if H is non-compact. The first half of the article elucidates general machinery to study discontinuous groups for G/H, followed by the most update and complete list of symmetric spaces with/without compact quotients. In the second half, as applica- tions of general theory, we prove: (i) there exists a 15 dimensional compact pseudo-Riemannian manifold of signature (7, 8) with constant curvature, (ii) there exists a compact quotient of the complex sphere of dimension 1, 3 and 7, and (iii) there exists a compact quotient of the tangential space form of signature (p, q) if and only if p is smaller than the Hurwitz-Radon number of q. Contents 1 Introduction 2 2 Examples 4 2.1 NotationandDefinition ....................... 4 2.2 Symmetric spaces with compact Clifford-Klein forms . 4 2.3 Para-Hermitiansymmetricspaces. 5 arXiv:math/0509543v1 [math.DG] 23 Sep 2005 2.4 Complex symmetric spaces GC/KC ................. 6 2.5 Spaceformproblem ......................... 8 2.6 Tangentspaceversionofspaceformproblem . 9 3 Methods 10 3.1 Generalized concept of properly discontinuous actions . 10 3.2 Criterionofpropernessforreductivegroups . 13 2000 Mathematics Subject Classification. Primary 22F30; Secondary 22E40, 53C30, 53C35, 57S30 Key words: discontinuous group, Clifford-Klein form, symmetric space, space form, pseudo- Riemannian manifold, discrete subgroup, uniform lattice, indefinite Clifford algebra 1 3.3 Construction of compact Clifford-Klein forms .
    [Show full text]
  • 3-Manifolds with Subgroups Z Z Z in Their Fundamental Groups
    Pacific Journal of Mathematics 3-MANIFOLDS WITH SUBGROUPS Z ⊕ Z ⊕ Z IN THEIR FUNDAMENTAL GROUPS ERHARD LUFT AND DENIS KARMEN SJERVE Vol. 114, No. 1 May 1984 PACIFIC JOURNAL OF MATHEMATICS Vol 114, No. 1, 1984 3-MANIFOLDS WITH SUBGROUPS ZΦZΦZ IN THEIR FUNDAMENTAL GROUPS E. LUFT AND D. SJERVE In this paper we characterize those 3-manifolds M3 satisfying ZΘZΘZC ^i(Λf). All such manifolds M arise in one of the following ways: (I) M = Mo # R, (II) M= Mo # R*, (III) M = Mo Uθ R*. Here 2 Λf0 is any 3-manifold in (I), (II) and any 3-manifold having P compo- nents in its boundary in (III). R is a flat space form and R* is obtained from R and some involution t: R -> R with fixed points, but only finitely many, as follows: if C,,..., Cn are disjoint 3-cells around the fixed points then R* is the 3-manifold obtained from (R - int(C, U UQ))/ί by identifying some pairs of projective planes in the boundary. 1. Introduction. In [1] it was shown that the only possible finitely generated abelian subgroups of the fundamental groups of 3-manifolds are Zn9 Z θ Z2, Z, Z θ Z and Z θ Z θ Z. The purpose of this paper is to 3 characterize all M satisfying ZΘZΘZC πx(M). To explain this characterization recall that the Bieberbach theorem (see Chapter 3 of [8]) implies that if M is a closed 3-dimensional flat space form then ZΘZΘZC πx(M). We let M,,... 9M6 denote the 6 compact connected orientable flat space forms in the order given on p.
    [Show full text]
  • Fundamental Groups of Manifolds with Positive Sectional Curvature And
    FUNDAMENTAL GROUPS OF MANIFOLDS WITH POSITIVE SECTIONAL CURVATURE AND TORUS SYMMETRY LEE KENNARD Abstract. In 1965, S.-S. Chern posed a question concerning the extent to which funda- mental groups of manifolds admitting positive sectional curvature look like spherical space form groups. The original question was answered in the negative by Shankar in 1998, but there are a number of positive results in the presence of symmetry. These classifications fall into categories according to the strength of their conclusions. We give an overview of these results in the case of torus symmetry and prove new results in each of these categories. 1. Introduction The sphere is the simplest example of a compact manifold. With its standard Riemannian metric, it has positive sectional curvature. Strikingly, in all odd dimensions except for 7 and 13, the spheres remain the only simply connected, compact manifolds known to admit pos- itively curved metrics. In view of this, it is not surprising that the realization problem for fundamental groups of positively curved manifolds remains open. As far as general obstruc- tions go, one only has the classical results of Synge and Bonnet–Myers, as well as Gromov’s universal bound on the number of generators (see [19]). Spherical space forms provide many examples of fundamental groups of positively curved manifolds (see Davis–Milgram [14], Hambleton [27], and Wolf [55]). However they form a very restrictive class of groups. In particular, they satisfy the properties that every abelian subgroup is cyclic and every involution is central. By analogy with Preissmann’s theorem, S.-S. Chern asked whether the first of these properties holds for the fundamental groups of positively curved manifolds (see [30, p.
    [Show full text]
  • Curvature Inequalities for Submanifolds of S-Space Form Najma Abdul Rehman Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus, Pakistan
    EUROPEAN JOURNAL OF PURE AND APPLIED MATHEMATICS Vol. 12, No. 4, 2019, 1811-1818 ISSN 1307-5543 { www.ejpam.com Published by New York Business Global Curvature Inequalities for Submanifolds of S-space form Najma Abdul Rehman Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus, Pakistan Abstract. In this paper we establish new results of squared mean curvature and Ricci curvature for the sub manifolds of S-space from that is the generalization of complex and contact structures. Obtained results are discussed for invariant, anti invariant and CR sub manifolds of S-space from. 2010 Mathematics Subject Classifications: 53C40, 53C25 Key Words and Phrases: Curvature, Sub manifolds, S-space form 1. Introduction One of the main and useful idea in submanifolds conjectures is to derive relationship among squared mean curvature and Ricci curvature of submanifolds, was explained by Chen [6], [7]. After this many authors derived Chen inequalities for complex and contact space forms [1], [11], [12] and on hyper surfaces of a Lorentzian manifold [9]. After the generalization of complex and contact space forms into S-space form [4], it is natural to study the inequalities of Ricci curvature for submanifolds of S-space forms. Geometry of S-space forms were studied by many authors i.e. [10], [13]. In this paper we find relations between squared mean curvature and Ricci curvature for the sub manifolds of S-space form and discuss this relation for invariant, anti invariant and CR sub manifolds of S-space form. After introduction, second section contains basics of S-space forms and submanifolds. Third section contains main results.
    [Show full text]
  • Integral Formulas and Hyperspheres in a Simply Connected Space Form
    proceedings of the american mathematical society Volume 88, Number 1, May 1983 INTEGRAL FORMULAS AND HYPERSPHERES IN A SIMPLY CONNECTED SPACE FORM IRL BIVENS Abstract. Let M" denote a connected compact hypersurface without boundary contained in Euclidean or hyperbolic n + 1 space or in an open hemisphere of S"+ '. We show that if two consecutive mean curvatures of M are constant then M is in fact a geodesic sphere. The proof uses the generalized Minkowski integral formulas for a hypersurface of a complete simply connected space form. These Minkowski formulas are derived from an integral formula for submanifolds in which the ambient Riemannian manifold M possesses a generalized position vector field; that is a vector field Y whose covariant derivative is at each point a multiple of the identity. In addition we prove that if M is complete and connected with the covariant derivative of Y exactly the identity at each point then M is isometric to Euclidean space. Let (M, g) denote a connected compact oriented n dimensional Riemannian manifold without boundary with h: M -» M an isometric immersion of M into an oriented n + p dimensional Riemannian manifold (M, g). If p = 1 we let ok denote the k th mean curvature of M. That is ok is equal to the k th elementary symmetric function of the principal curvatures of M divided by ( k ). Theorem 1. Suppose p — 1 and assume ok, ak+x are constant for some k = 1,2,...,« - 1. (a) If M = R"+ ' with the standard metric then h embeds M as an n-sphere in R"+ .
    [Show full text]
  • Metric Foundations of Geometry. I
    METRIC FOUNDATIONS OF GEOMETRY. I BY GARRETT BIRKHOFF 1. Introduction. It is shown below by elementary methods that «-dimen- sional Euclidean, spherical, and hyperbolic geometry can be characterized by the following postulates. Postulate I. Space is metric (in the sense of Fréchet). Postulate II. Through any two points a line segment can be drawn, locally uniquely. Postulate III. Any isometry between subsets of space can be extended to a self-isometry of all space. The outline of the proof has been presented elsewhere (G. Birkhoff [2 ]('))• Earlier, H. Busemann [l] had shown that if, further, exactly one straight line connected any two points, and if bounded sets were compact, then the space was Euclidean or hyperbolic; moreover for this, Postulate III need only be assumed for triples of points. Later, Busemann extended the result to elliptic space, assuming that exactly one geodesic connected any two points, that bounded sets were compact, and that Postulate III held locally for triples of points (H. Busemann [2, p. 186]). We recall Lie's celebrated proof that any Riemannian variety having local free mobility is locally Euclidean, spherical, or hyperbolic. Since our Postu- late II, unlike Busemann's straight line postulates, holds in any Riemannian variety in which bounded sets are compact, our result may be regarded as freeing Lie's theorem for the first time from its analytical hypotheses and its local character. What is more important, we use direct geometric arguments, where Lie used sophisticated analytical and algebraic methods(2). Our methods are also far more direct and elementary than those of Busemann, who relies on a deep theorem of Hurewicz.
    [Show full text]
  • Symmetries of Eschenburg Spaces and the Chern Problem
    SYMMETRIES OF ESCHENBURG SPACES AND THE CHERN PROBLEM KARSTEN GROVE, KRISHNAN SHANKAR, WOLFGANG ZILLER Dedicated to the memory of S. S. Chern. To advance our basic knowledge of manifolds with positive (sectional) curvature it is essential to search for new examples, and to get a deeper understanding of the known ones. Although any positively curved manifold can be perturbed so as to have trivial isometry group, it is natural to look for, and understand the most symmetric ones, as in the case of homogeneous spaces. In addition to the compact rank one symmetric spaces, the complete list (see [BB]) of simply connected homogeneous manifolds of positive curvature consists of the Berger spaces B7 and B13 [Be], the Wallach spaces W 6,W 12 and W 24 [Wa], and the infinite class of so- called Aloff–Wallach spaces, A7 [AW]. Their full isometry groups were determined in [Sh2], and this knowledge provided new basic information about possible fundamental groups of positively curved manifolds, and in particular to counter-examples of the so-called Chern conjecture (see [Sh1] and [GSh, Ba2]), which states that every abelian subgroup of the fundamental group is cyclic. Our purpose here is to begin a systematic analysis of the isometry groups of the remaining known manifolds of positive curvature, i.e., of the so-called Eschenburg spaces, E7 [Es1, Es2] (plus one in dimension 6) and the Bazaikin spaces, B13 [Ba1], with an emphasis on the former. In particular, we completely determine the identity component of the isometry group of any positively curved Eschenburg space. A member of E is a so-called bi-quotient of SU(3) by a circle: E = diag(zk1 , zk2 , zk3 )\ SU(3)/ diag(zl1 , zl2 , zl3 )−1, |z| = 1 P P with ki = li.
    [Show full text]
  • Stability on S-Space Form
    Indian J. Pure Appl. Math., 50(4): 1087-1096, December 2019 °c Indian National Science Academy DOI: 10.1007/s13226-019-0375-y STABILITY ON S-SPACE FORM Najma Abdul Rehman Department of Mathematics, COMSATS University Islamabad, Sahiwal Campus, 57000, Pakistan e-mail: najma [email protected] (Received 6 February 2017; accepted 27 November 2018) In this paper, the stability of the identity map on a compact S-space form is studied. Results related to stability of holomorphic maps on compact domain of S-space form are also discussed. Key words : Harmonic maps; stability; holomorphic maps; S-space form; Ka¨hler manifold. 1. INTRODUCTION Harmonic maps theory takes ideas from both Riemannian geometry and analysis. There are many attractive results about harmonic maps on complex manifolds (see [12, 20]). In the analogy to the complex case, in the last decade harmonic maps on almost contact metric manifolds were studied [3, 4, 6, 7, 10]. The identity map of a compact Riemannian manifold is a trivial example of a harmonic map but in this case, the theory of the second variation is much complicated and interesting. For example, the stability of the identity map on Einstein manifolds is related to the first eigenvalue of the Laplacian acting on functions [18]. In [19] and [14] we find classifications of compact simply connected irreducible Riemannian symmetric spaces for which the identity map is unstable. By a well known result, the identity map on the euclidean sphere S2n+1 is unstable [18]. More generally, Gherghe, Ianus and Pastore have studied the stability of the identity map on compact Sasakian manifolds with constant '¡sectional curvature [10].
    [Show full text]
  • CHAPTER 6: RICCI FLOW Contents 1. the Hamilton–Perelman Program
    CHAPTER 6: RICCI FLOW DANNY CALEGARI Abstract. These are notes on Ricci Flow on 3-Manifolds after Hamilton and Perelman, which are being transformed into Chapter 6 of a book on 3-Manifolds. These notes are based on a graduate course taught at the University of Chicago in Fall 2019. Contents 1. The Hamilton–Perelman program1 2. Mean curvature flow: a comparison8 3. Curvature evolution and pinching 15 4. Singularities and Limits 33 5. Perelman’s monotone functionals 41 6. The Geometrization Conjecture 54 7. Acknowledgments 61 References 61 1. The Hamilton–Perelman program In this section we give a very informal overview of the Hamilton–Perelman program proving the Poincaré Conjecture and the Geometrization Conjecture for 3-manifolds. 1.1. What is Ricci flow? There’s lots of different ways to answer this question, depending on the point you want to emphasize. There’s no getting around the precision and economy of a formula, but for now let’s see how far we can get with mostly words. First of all, what is curvature? To be differentiable is to have a good linear approximation at each point: the derivative. To be smooth is for successive derivatives to be themselves differentiable; for example, the deviation of a smooth function from its derivative has a good quadratic approximation at each point: the Hessian. A Riemannian manifold is a space which is Euclidean (i.e. flat) to first order. Riemannian manifolds are smooth, so there is a well-defined second order deviation from flatness, and that’s Curvature. In Euclidean space of dimension n a ball of radius r has volume rn times a (dimension dependent) constant.
    [Show full text]
  • Ricci Flow and Diffeomorphism Groups of 3-Manifolds
    RICCI FLOW AND DIFFEOMORPHISM GROUPS OF 3-MANIFOLDS RICHARD H. BAMLER AND BRUCE KLEINER Abstract. We complete the proof of the Generalized Smale Conjecture, apart from the case of RP 3, and give a new proof of Gabai's theorem for hyperbolic 3- manifolds. We use an approach based on Ricci flow through singularities, which applies uniformly to spherical space forms other than S3 and RP 3 and hyperbolic manifolds, to prove that the moduli space of metrics of constant sectional curvature is contractible. As a corollary, for such a 3-manifold X, the inclusion Isom(X; g) ! Diff(X) is a homotopy equivalence for any Riemannian metric g of constant sectional curvature. Contents 1. Introduction 1 2. Preliminaries 6 3. The canonical limiting constant curvature metric 16 4. Extending constant curvature metrics 19 5. Proof of Theorem 1.2 for spherical space forms 25 6. Proof of Theorem 1.2 for hyperbolic manifolds 26 References 28 1. Introduction Let X be a compact connected smooth 3-manifold. We let Diff(X) and Met(X) denote the group of smooth diffeomorphisms of X, and the set of Riemannian metrics on X, respectively, equipped with their C1-topologies. Our focus in this paper will be on the following conjecture: Conjecture 1.1 (Generalized Smale Conjecture [Sma61, Gab01, HKMR12]). If g is a Riemannian metric of constant sectional curvature ±1 on X, then the inclusion Isom(X; g) ,! Diff(X) is a homotopy equivalence. Date: December 15, 2017. The first author was supported by a Sloan Research Fellowship and NSF grant DMS-1611906.
    [Show full text]
  • The Clifford-Klein Space Forms of Indefinite Metric Author(S): Joseph A
    Annals of Mathematics The Clifford-Klein Space Forms of Indefinite Metric Author(s): Joseph A. Wolf Reviewed work(s): Source: The Annals of Mathematics, Second Series, Vol. 75, No. 1 (Jan., 1962), pp. 77-80 Published by: Annals of Mathematics Stable URL: http://www.jstor.org/stable/1970420 . Accessed: 03/05/2012 14:54 Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at . http://www.jstor.org/page/info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact [email protected]. Annals of Mathematics is collaborating with JSTOR to digitize, preserve and extend access to The Annals of Mathematics. http://www.jstor.org ANNAL!S OF MIATHEMATICS Vol. 75, No. 1, January, 1962 Printed in Japan THE CLIFFORD-KLEIN SPACE FORMS OF INDEFINITE METRIC BY JOSEPH A. WOLF* (Received February 24, 1961) 1. Introduction The sphericalspace formproblem of Clifford-Kleinis the classification problemfor complete connected riemannian manifolds of constantpositive curvature. E. Calabi and L. Markushave recentlyconsidered the classi- ficationproblem for completeconnected Lorenz n-manifoldsM," of con- stant positive curvature,and have [1; Theorems2 and 3] reducedit to the sphericalspace formproblem for riemannian (n - 1)-manifolds,when n ? 3. We will extendtheir ideas to moregeneral signaturesof metric, and reduce the classificationproblem for complete connectedpseudo- riemanniann-manifolds M,, of constantpositive curvature, and with s # n - 1 and 2s < n, to the sphericalspace formproblem for riemannian (n - s)-manifolds.
    [Show full text]
  • Unit 12. Curvature ======
    Unit 12. Curvature ========================================================================================== ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- Curvature operator, curvature tensor, Bianchi identities, Riemann-Christoffel tensor, symmetry properties of the Riemann-Christoffel tensor, sectional curvature, Schur’s theorem, space forms, Ricci tensor, Ricci curvature, scalar curvature, curvature tensor of a hypersurface. ------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------------- If D is an affine connection on a manifold M, then we may consider the operator R(X,Y) = [DX,D Y]- D [X,Y]:X(M)----------L X(M), where [DXYXYYX,D ]=D qD -D qD is the usual commutator of operators. The mapping that assigns to the vector fields X,Y the operator R(X,Y) is called the ------------------------------------------------------------------------------------------curvature operator of the connection.
    [Show full text]