DOCSLIB.ORG

GEOMETRIC INTERPRETATIONS of CURVATURE Contents 1. Notation and Summation Conventions 1 2. Affine Connections 1 3. Parallel Tran

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
GEOMETRIC INTERPRETATIONS of CURVATURE Contents 1. Notation and Summation Conventions 1 2. Affine Connections 1 3. Parallel Tran GEOMETRIC INTERPRETATIONS OF CURVATURE ZHENGQU WAN Abstract. This is an expository paper on geometric meaning of various kinds of curvature on a Riemann manifold. Contents 1. Notation and Summation Conventions 1 2. Affine Connections 1 3. Parallel Transport 3 4. Geodesics and the Exponential Map 4 5. Riemannian Curvature Tensor 5 6. Taylor Expansion of the Metric in Normal Coordinates and the Geometric Interpretation of Ricci and Scalar Curvature 9 Acknowledgments 13 References 13 1. Notation and Summation Conventions We assume knowledge of the basic theory of smooth manifolds, vector fields and tensors. We will assume all manifolds are smooth, i.e. C1, second countable and Hausdorff. All functions, curves and vector fields will also be smooth unless otherwise stated. Einstein summation convention will be adopted in this paper. In some cases, the index types on either side of an equation will not match and @ so a summation will be needed. The tangent vector field @xi induced by local i coordinates (x ) will be denoted as @i. 2. Affine Connections Riemann curvature is a measure of the noncommutativity of parallel transporta- tion of tangent vectors. To define parallel transport, we need the notion of affine connections. Definition 2.1. Let M be an n-dimensional manifold. An affine connection, or connection, is a map r : X(M) × X(M) ! X(M), where X(M) denotes the space of smooth vector fields, such that for vector fields V1;V2; V; W1;W2 2 X(M) and function f : M! R, (1) r(fV1 + V2;W ) = fr(V1;W ) + r(V2;W ), (2) r(V; aW1 + W2) = ar(V; W1) + r(V; W2), for all a 2 R. (3) r(V; fW ) = V (f)W + fr(V; W ). We write r(V; W ) = rV W . 1 2 ZHENGQU WAN Theorem 2.2. For fixed W , (rV W )jp only depends on V jp. Proof. It suffices to show if V jp = 0 then (rU W )jp = 0. The proof relies on the following lemma: if V jp = 0 then there exists smooth scalar fields fk and vector ~ P ~ fields Vk such that fk(p) = 0 and V = k fkVk. This fact is easily proven using P ~ partitions of unity. Using this lemma, we then write V = k fkVk where fk(p) = 0. Then X (rP ~ W )jp = fk(p)(r ~ W )jp = 0: k fkVk Vk k Remark 2.3. Since rV W depends on V pointwise, we may write rvW := rV W jp where v = Vp 2 TpM. We also note that rV W jp depends on the local values of W , that is, if W1 = W2 in some neighborhood of p, then rvW1jp = rvW2jp for all v 2 TpM. We now give an expression for the connection r in local coordinates (xi) on an j i open set U. Suppose we have two vector fields W = w @j and V = v @i on U. Then @vi r V = wj @ + wjvir @ : W @xj i @j i k We define functions Γij : U ! R, for 1 ≤ i; j; k ≤ n, by k r@j @i = Γij@k: These functions are called the Christoffel symbols and depend on the coordinate system used to define them. Since covariant differentiation takes values in X(M), we see immediately that the Christoffel symbols are smooth. We now show existence of a global affine connection on a manifold M. Theorem 2.4. For any manifold M, there exists an affine connection on M. Proof. Let fUαgα be an atlas of M and let fραgα be the associated partition of unity. On each open set Uα, we have a connection rα given in the coordinates on k P Uα by Γij = 0. We define r by rV W = α(rα)ρα·V (ρα · W ). Note that at each point p, rjp is a finite sum. Some straightforward computations will show that r satisfies the three properties in Theorem 2.1 and rV W is smooth when V; W are smooth. Remark 2.5. The affine connection r on a manifold need not be unique. If we k vary the Christoffel symbols Γij in each of the charts Uα of the proof, we may get different affine connections. Remark 2.6. We know that rwV depends on W locally, however we can further show that if η is a smooth curve such that η(0) = p and η0(0) = w, then the value of rwV only depends on behavior of V on the curve η. Thus, rwV is well-defined even if V is only defined along a curve through p and tangent to w. Definition 2.7. Let η : I !M be a smooth curve. A vector field V along η is a smooth map V : I ! T M such that for each t 2 I, V (t) 2 Tη(t)M. Remark 2.6 above says that for a vector v, the map W 7! rvW is defined on vector fields W along curves η such that η(0) = p and η0(0) = w. Definition 2.8. A vector field V along a curve η is parallel if rη_(t)V = 0 for all t. GEOMETRIC INTERPRETATIONS OF CURVATURE 3 Theorem 2.9. Let η : [0; 1] !M be a smooth curve and let v 2 Tη(0)M. Then, there exists a unique parallel vector field V along η such that V (0) = v. Proof. First, we consider the case when η is contained in a local coordinate chart. k In local coordinates, V (t) = V (t)@k satisfies the system of ODEs dV k + V iη_jΓk = 0 for k = 1; : : : ; n dt ij with initial conditions V k(0) = vk. Observe that the map k k j j (V ; t) 7! −Γij(η(t))V η_ (t) k k j j is continuous; for fixed t, the map (V ) 7! −ΓijV η_ is linear and thus globally Lipchitz continuous in V k. Picard-Lindelof theorem tells us that this equation has a unique solution V (t). For general η, we let 0 = t0 < ··· < tN = 1 be a partition of [0; 1] such that, for each i, ηj[ti−1;ti] is contained in a coordinate chart Ui. The previous paragraph implies that there is a parallel vector field V along ηj[0;t1] such that V (0) = v. If ~ we have a parallel vector field V along ηj[0;ti], we obtain a vector field V along ~ ηj[ti;ti+1] such that V (ti) = V (ti), again using the previous paragraph. Then, V ~ extends to a vector field along ηj[0;ti+1] by defining V (t) = V (t) for t 2 [ti; ti+1]. By induction, we get a vector field V which is parallel along all of η. 3. Parallel Transport One problem on general manifolds is, unlike Euclidean spaces, if p; q 2 M are dis- tinct points, then there is no natural identification TpM = TqM. In this section, we illustrate how we may construct such an isomorphism, called parallel trans- portation, using the notion of parallel vector fields. However, this isomorphism will depend on a choice of path between p and q. Definition 3.1. Let η : [0; 1] !M be a curve connecting two points p; q 2 M, i.e. η(0) = p and η(1) = q. Let v 2 TpM and let V denote the parallel vector field along η ensured by Theorem 2.9. The vector V (1) is called the parallel transport of v along η and the map v 7! V (1) is denoted Tη(0)!η(1) or Tp!q or by T if the path η is clear from the context. Remark 3.2. Since the equation defining parallel transportation is a linear ODE, for any path η from p to q, we have Tp!q(av + w) = aTp!qv + Tp!qw. From the definition of parallel transport we see that Tp!q is the inverse of Tq!p. Thus Tp!q is an isomorphism from TpM to TqM. Remark 3.3. Let η be a curve and p = η(0). If fv1;:::; vng is a basis of TpM, then we can extend vi to parallel vector fields Ei(t) along η. Since parallel transportation is invertible, fE1(t);:::;En(t)g forms a basis of Tη(t)M at each t. We have defined parallel transport in terms of the affine connection. The fol- lowing theorem shows that we may define the affine connection in terms of parallel transport. Theorem 3.4. Suppose we have a smooth vector field V and a smooth curve η 1 such that η_(0) = v. Then lim!0+ Tη()!η(0)V () − V (0) = rη_ (0)V . 4 ZHENGQU WAN Proof. Let E1;:::;En be parallel vector fields along η forming a basis at each point. k k Write V (t) = V (t)Ek(t). Then, Tη()!η(0)V () = V ()Ek(0), since Tη()!η(0) is a linear map. Then, 1 1 (T V () − V (0)) = (V k() − V k(0))E (0): η()!η(0) k d k Taking the limit ! 0 gives us dt jt=0(V )(t)Ek(0). Since Ek is parallel, we also have that 0 k k 0 k rη0(0)V (t)jt=0 = η (0)(V (t))Ek(0) + V (0)rη0(0)Ek(t) = η (0)(V (t))Ek(0): d k This is, by definition dt jt=0(V )(t)Ek(0), so the two quantities are the same. Remark 3.5. Note that this theorem states that the affine connection is equal to to the limit of a difference quotient.
Load more

Recommended publications
  • Scalar Curvature and Geometrization Conjectures for 3-Manifolds
    Comparison Geometry MSRI Publications Volume 30, 1997 Scalar Curvature and Geometrization Conjectures for 3-Manifolds MICHAEL T. ANDERSON Abstract. We first summarize very briefly the topology of 3-manifolds and the approach of Thurston towards their geometrization. After dis- cussing some general properties of curvature functionals on the space of metrics, we formulate and discuss three conjectures that imply Thurston’s Geometrization Conjecture for closed oriented 3-manifolds. The final two sections present evidence for the validity of these conjectures and outline an approach toward their proof. Introduction In the late seventies and early eighties Thurston proved a number of very re- markable results on the existence of geometric structures on 3-manifolds. These results provide strong support for the profound conjecture, formulated by Thur- ston, that every compact 3-manifold admits a canonical decomposition into do- mains, each of which has a canonical geometric structure. For simplicity, we state the conjecture only for closed, oriented 3-manifolds. Geometrization Conjecture [Thurston 1982]. Let M be a closed , oriented, 2 prime 3-manifold. Then there is a finite collection of disjoint, embedded tori Ti 2 in M, such that each component of the complement M r Ti admits a geometric structure, i.e., a complete, locally homogeneous RiemannianS metric. A more detailed description of the conjecture and the terminology will be given in Section 1. A complete Riemannian manifold N is locally homogeneous if the universal cover N˜ is a complete homogenous manifold, that is, if the isometry group Isom N˜ acts transitively on N˜. It follows that N is isometric to N=˜ Γ, where Γ is a discrete subgroup of Isom N˜, which acts freely and properly discontinuously on N˜.
    [Show full text]
  • Connections on Bundles Md
    Dhaka Univ. J. Sci. 60(2): 191-195, 2012 (July) Connections on Bundles Md. Showkat Ali, Md. Mirazul Islam, Farzana Nasrin, Md. Abu Hanif Sarkar and Tanzia Zerin Khan Department of Mathematics, University of Dhaka, Dhaka 1000, Bangladesh, Email: [email protected] Received on 25. 05. 2011.Accepted for Publication on 15. 12. 2011 Abstract This paper is a survey of the basic theory of connection on bundles. A connection on tangent bundle , is called an affine connection on an -dimensional smooth manifold . By the general discussion of affine connection on vector bundles that necessarily exists on which is compatible with tensors. I. Introduction = < , > (2) In order to differentiate sections of a vector bundle [5] or where <, > represents the pairing between and ∗. vector fields on a manifold we need to introduce a Then is a section of , called the absolute differential structure called the connection on a vector bundle. For quotient or the covariant derivative of the section along . example, an affine connection is a structure attached to a differentiable manifold so that we can differentiate its Theorem 1. A connection always exists on a vector bundle. tensor fields. We first introduce the general theorem of Proof. Choose a coordinate covering { }∈ of . Since connections on vector bundles. Then we study the tangent vector bundles are trivial locally, we may assume that there is bundle. is a -dimensional vector bundle determine local frame field for any . By the local structure of intrinsically by the differentiable structure [8] of an - connections, we need only construct a × matrix on dimensional smooth manifold . each such that the matrices satisfy II.
    [Show full text]
  • Laplacians in Geometric Analysis
    LAPLACIANS IN GEOMETRIC ANALYSIS Syafiq Johar syafi[email protected] Contents 1 Trace Laplacian 1 1.1 Connections on Vector Bundles . .1 1.2 Local and Explicit Expressions . .2 1.3 Second Covariant Derivative . .3 1.4 Curvatures on Vector Bundles . .4 1.5 Trace Laplacian . .5 2 Harmonic Functions 6 2.1 Gradient and Divergence Operators . .7 2.2 Laplace-Beltrami Operator . .7 2.3 Harmonic Functions . .8 2.4 Harmonic Maps . .8 3 Hodge Laplacian 9 3.1 Exterior Derivatives . .9 3.2 Hodge Duals . 10 3.3 Hodge Laplacian . 12 4 Hodge Decomposition 13 4.1 De Rham Cohomology . 13 4.2 Hodge Decomposition Theorem . 14 5 Weitzenb¨ock and B¨ochner Formulas 15 5.1 Weitzenb¨ock Formula . 15 5.1.1 0-forms . 15 5.1.2 k-forms . 15 5.2 B¨ochner Formula . 17 1 Trace Laplacian In this section, we are going to present a notion of Laplacian that is regularly used in differential geometry, namely the trace Laplacian (also called the rough Laplacian or connection Laplacian). We recall the definition of connection on vector bundles which allows us to take the directional derivative of vector bundles. 1.1 Connections on Vector Bundles Definition 1.1 (Connection). Let M be a differentiable manifold and E a vector bundle over M. A connection or covariant derivative at a point p 2 M is a map D : Γ(E) ! Γ(T ∗M ⊗ E) 1 with the properties for any V; W 2 TpM; σ; τ 2 Γ(E) and f 2 C (M), we have that DV σ 2 Ep with the following properties: 1.
    [Show full text]
  • The Simplicial Ricci Tensor 2
    The Simplicial Ricci Tensor Paul M. Alsing1, Jonathan R. McDonald 1,2 & Warner A. Miller3 1Information Directorate, Air Force Research Laboratory, Rome, New York 13441 2Insitut f¨ur Angewandte Mathematik, Friedrich-Schiller-Universit¨at-Jena, 07743 Jena, Germany 3Department of Physics, Florida Atlantic University, Boca Raton, FL 33431 E-mail: [email protected] Abstract. The Ricci tensor (Ric) is fundamental to Einstein’s geometric theory of gravitation. The 3-dimensional Ric of a spacelike surface vanishes at the moment of time symmetry for vacuum spacetimes. The 4-dimensional Ric is the Einstein tensor for such spacetimes. More recently the Ric was used by Hamilton to define a non-linear, diffusive Ricci flow (RF) that was fundamental to Perelman’s proof of the Poincar`e conjecture. Analytic applications of RF can be found in many fields including general relativity and mathematics. Numerically it has been applied broadly to communication networks, medical physics, computer design and more. In this paper, we use Regge calculus (RC) to provide the first geometric discretization of the Ric. This result is fundamental for higher-dimensional generalizations of discrete RF. We construct this tensor on both the simplicial lattice and its dual and prove their equivalence. We show that the Ric is an edge-based weighted average of deficit divided by an edge-based weighted average of dual area – an expression similar to the vertex-based weighted average of the scalar curvature reported recently. We use this Ric in a third and independent geometric derivation of the RC Einstein tensor in arbitrary dimension. arXiv:1107.2458v1 [gr-qc] 13 Jul 2011 The Simplicial Ricci Tensor 2 1.
    [Show full text]
  • Selected Papers on Teleparallelism Ii
    SELECTED PAPERS ON TELEPARALLELISM Edited and translated by D. H. Delphenich Table of contents Page Introduction ……………………………………………………………………… 1 1. The unification of gravitation and electromagnetism 1 2. The geometry of parallelizable manifold 7 3. The field equations 20 4. The topology of parallelizability 24 5. Teleparallelism and the Dirac equation 28 6. Singular teleparallelism 29 References ……………………………………………………………………….. 33 Translations and time line 1928: A. Einstein, “Riemannian geometry, while maintaining the notion of teleparallelism ,” Sitzber. Preuss. Akad. Wiss. 17 (1928), 217- 221………………………………………………………………………………. 35 (Received on June 7) A. Einstein, “A new possibility for a unified field theory of gravitation and electromagnetism” Sitzber. Preuss. Akad. Wiss. 17 (1928), 224-227………… 42 (Received on June 14) R. Weitzenböck, “Differential invariants in EINSTEIN’s theory of teleparallelism,” Sitzber. Preuss. Akad. Wiss. 17 (1928), 466-474……………… 46 (Received on Oct 18) 1929: E. Bortolotti , “ Stars of congruences and absolute parallelism: Geometric basis for a recent theory of Einstein ,” Rend. Reale Acc. dei Lincei 9 (1929), 530- 538...…………………………………………………………………………….. 56 R. Zaycoff, “On the foundations of a new field theory of A. Einstein,” Zeit. Phys. 53 (1929), 719-728…………………………………………………............ 64 (Received on January 13) Hans Reichenbach, “On the classification of the new Einstein Ansatz on gravitation and electricity,” Zeit. Phys. 53 (1929), 683-689…………………….. 76 (Received on January 22) Selected papers on teleparallelism ii A. Einstein, “On unified field theory,” Sitzber. Preuss. Akad. Wiss. 18 (1929), 2-7……………………………………………………………………………….. 82 (Received on Jan 30) R. Zaycoff, “On the foundations of a new field theory of A. Einstein; (Second part),” Zeit. Phys. 54 (1929), 590-593…………………………………………… 89 (Received on March 4) R.
    [Show full text]
  • 2. Chern Connections and Chern Curvatures1
    1 2. Chern connections and Chern curvatures1 Let V be a complex vector space with dimC V = n. A hermitian metric h on V is h : V £ V ¡¡! C such that h(av; bu) = abh(v; u) h(a1v1 + a2v2; u) = a1h(v1; u) + a2h(v2; u) h(v; u) = h(u; v) h(u; u) > 0; u 6= 0 where v; v1; v2; u 2 V and a; b; a1; a2 2 C. If we ¯x a basis feig of V , and set hij = h(ei; ej) then ¤ ¤ ¤ ¤ h = hijei ­ ej 2 V ­ V ¤ ¤ ¤ ¤ where ei 2 V is the dual of ei and ei 2 V is the conjugate dual of ei, i.e. X ¤ ei ( ajej) = ai It is obvious that (hij) is a hermitian positive matrix. De¯nition 0.1. A complex vector bundle E is said to be hermitian if there is a positive de¯nite hermitian tensor h on E. r Let ' : EjU ¡¡! U £ C be a trivilization and e = (e1; ¢ ¢ ¢ ; er) be the corresponding frame. The r hermitian metric h is represented by a positive hermitian matrix (hij) 2 ¡(­; EndC ) such that hei(x); ej(x)i = hij(x); x 2 U Then hermitian metric on the chart (U; ') could be written as X ¤ ¤ h = hijei ­ ej For example, there are two charts (U; ') and (V; Ã). We set g = à ± '¡1 :(U \ V ) £ Cr ¡¡! (U \ V ) £ Cr and g is represented by matrix (gij). On U \ V , we have X X X ¡1 ¡1 ¡1 ¡1 ¡1 ei(x) = ' (x; "i) = à ± à ± ' (x; "i) = à (x; gij"j) = gijà (x; "j) = gije~j(x) j j For the metric X ~ hij = hei(x); ej(x)i = hgike~k(x); gjle~l(x)i = gikhklgjl k;l that is h = g ¢ h~ ¢ g¤ 12008.04.30 If there are some errors, please contact to: [email protected] 2 Example 0.2 (Fubini-Study metric on holomorphic tangent bundle T 1;0Pn).
    [Show full text]
  • “Geodesic Principle” in General Relativity∗
    A Remark About the “Geodesic Principle” in General Relativity∗ Version 3.0 David B. Malament Department of Logic and Philosophy of Science 3151 Social Science Plaza University of California, Irvine Irvine, CA 92697-5100 [email protected] 1 Introduction General relativity incorporates a number of basic principles that correlate space- time structure with physical objects and processes. Among them is the Geodesic Principle: Free massive point particles traverse timelike geodesics. One can think of it as a relativistic version of Newton’s first law of motion. It is often claimed that the geodesic principle can be recovered as a theorem in general relativity. Indeed, it is claimed that it is a consequence of Einstein’s ∗I am grateful to Robert Geroch for giving me the basic idea for the counterexample (proposition 3.2) that is the principal point of interest in this note. Thanks also to Harvey Brown, Erik Curiel, John Earman, David Garfinkle, John Manchak, Wayne Myrvold, John Norton, and Jim Weatherall for comments on an earlier draft. 1 ab equation (or of the conservation principle ∇aT = 0 that is, itself, a conse- quence of that equation). These claims are certainly correct, but it may be worth drawing attention to one small qualification. Though the geodesic prin- ciple can be recovered as theorem in general relativity, it is not a consequence of Einstein’s equation (or the conservation principle) alone. Other assumptions are needed to drive the theorems in question. One needs to put more in if one is to get the geodesic principle out. My goal in this short note is to make this claim precise (i.e., that other assumptions are needed).
    [Show full text]
  • Parallel Transport Along Seifert Manifolds and Fractional Monodromy Martynchuk, N.; Efstathiou, K
    University of Groningen Parallel Transport along Seifert Manifolds and Fractional Monodromy Martynchuk, N.; Efstathiou, K. Published in: Communications in Mathematical Physics DOI: 10.1007/s00220-017-2988-5 IMPORTANT NOTE: You are advised to consult the publisher's version (publisher's PDF) if you wish to cite from it. Please check the document version below. Document Version Publisher's PDF, also known as Version of record Publication date: 2017 Link to publication in University of Groningen/UMCG research database Citation for published version (APA): Martynchuk, N., & Efstathiou, K. (2017). Parallel Transport along Seifert Manifolds and Fractional Monodromy. Communications in Mathematical Physics, 356(2), 427-449. https://doi.org/10.1007/s00220- 017-2988-5 Copyright Other than for strictly personal use, it is not permitted to download or to forward/distribute the text or part of it without the consent of the author(s) and/or copyright holder(s), unless the work is under an open content license (like Creative Commons). Take-down policy If you believe that this document breaches copyright please contact us providing details, and we will remove access to the work immediately and investigate your claim. Downloaded from the University of Groningen/UMCG research database (Pure): http://www.rug.nl/research/portal. For technical reasons the number of authors shown on this cover page is limited to 10 maximum. Download date: 27-09-2021 Commun. Math. Phys. 356, 427–449 (2017) Communications in Digital Object Identifier (DOI) 10.1007/s00220-017-2988-5 Mathematical Physics Parallel Transport Along Seifert Manifolds and Fractional Monodromy N. Martynchuk , K.
    [Show full text]
  • Curvature Tensors in a 4D Riemann–Cartan Space: Irreducible Decompositions and Superenergy
    Curvature tensors in a 4D Riemann–Cartan space: Irreducible decompositions and superenergy Jens Boos and Friedrich W. Hehl [email protected] [email protected]"oeln.de University of Alberta University of Cologne & University of Missouri (uesday, %ugust 29, 17:0. Geometric Foundations of /ravity in (artu Institute of 0hysics, University of (artu) Estonia Geometric Foundations of /ravity Geometric Foundations of /auge Theory Geometric Foundations of /auge Theory ↔ Gravity The ingredients o$ gauge theory: the e2ample o$ electrodynamics ,3,. The ingredients o$ gauge theory: the e2ample o$ electrodynamics 0henomenological Ma24ell: redundancy conserved e2ternal current 5 ,3,. The ingredients o$ gauge theory: the e2ample o$ electrodynamics 0henomenological Ma24ell: Complex spinor 6eld: redundancy invariance conserved e2ternal current 5 conserved #7,8 current ,3,. The ingredients o$ gauge theory: the e2ample o$ electrodynamics 0henomenological Ma24ell: Complex spinor 6eld: redundancy invariance conserved e2ternal current 5 conserved #7,8 current Complete, gauge-theoretical description: 9 local #7,) invariance ,3,. The ingredients o$ gauge theory: the e2ample o$ electrodynamics 0henomenological Ma24ell: iers Complex spinor 6eld: rce carr ry of fo mic theo rrent rosco rnal cu m pic en exte att desc gredundancyiv er; N ript oet ion o conserved e2ternal current 5 invariance her f curr conserved #7,8 current e n t s Complete, gauge-theoretical description: gauge theory = complete description of matter and 9 local #7,) invariance how it interacts via gauge bosons ,3,. Curvature tensors electrodynamics :ang–Mills theory /eneral Relativity 0oincaré gauge theory *3,. Curvature tensors electrodynamics :ang–Mills theory /eneral Relativity 0oincaré gauge theory *3,. Curvature tensors electrodynamics :ang–Mills theory /eneral Relativity 0oincar; gauge theory *3,.
    [Show full text]
  • NON-EXISTENCE of METRIC on Tn with POSITIVE SCALAR
    NON-EXISTENCE OF METRIC ON T n WITH POSITIVE SCALAR CURVATURE CHAO LI Abstract. In this note we present Gromov-Lawson's result on the non-existence of metric on T n with positive scalar curvature. Historical results Scalar curvature is one of the simplest invariants of a Riemannian manifold. In general dimensions, this function (the average of all sectional curvatures at a point) is a weak measure of the local geometry, hence it's suspicious that it has no relation to the global topology of the manifold. In fact, a result of Kazdan-Warner in 1975 states that on a compact manifold of dimension ≥ 3, every smooth function which is negative somewhere, is the scalar curvature of some Riemannian metric. However people know that there are manifolds which carry no metric whose scalar scalar curvature is everywhere positive. The first examples of such manifolds were given in 1962 by Lichnerowicz. It is known that if X is a compact spin manifold and A^ 6= 0 then by Lichnerowicz formula (which will be discussed later) then X doesn't carry any metric with everywhere positive scalar curvature. Note that spin assumption is essential here, since the complex projective plane has positive sectional curvature and non-zero A^-genus. Despite these impressive results, one simple question remained open: Can the torus T n, n ≥ 3, carry a metric of positive scalar curvature? This question was settled for n ≤ 7 by R. Schoen and S. T. Yau. Their method involves the existence of smooth solution of Plateau problem, so the dimension restriction n ≤ 7 is essential.
    [Show full text]
  • Riemannian Geometry and Multilinear Tensors with Vector Fields on Manifolds Md
    International Journal of Scientific & Engineering Research, Volume 5, Issue 9, September-2014 157 ISSN 2229-5518 Riemannian Geometry and Multilinear Tensors with Vector Fields on Manifolds Md. Abdul Halim Sajal Saha Md Shafiqul Islam Abstract-In the paper some aspects of Riemannian manifolds, pseudo-Riemannian manifolds, Lorentz manifolds, Riemannian metrics, affine connections, parallel transport, curvature tensors, torsion tensors, killing vector fields, conformal killing vector fields are focused. The purpose of this paper is to develop the theory of manifolds equipped with Riemannian metric. I have developed some theorems on torsion and Riemannian curvature tensors using affine connection. A Theorem 1.20 named “Fundamental Theorem of Pseudo-Riemannian Geometry” has been established on Riemannian geometry using tensors with metric. The main tools used in the theorem of pseudo Riemannian are tensors fields defined on a Riemannian manifold. Keywords: Riemannian manifolds, pseudo-Riemannian manifolds, Lorentz manifolds, Riemannian metrics, affine connections, parallel transport, curvature tensors, torsion tensors, killing vector fields, conformal killing vector fields. —————————— —————————— I. Introduction (c) { } is a family of open sets which covers , that is, 푖 = . Riemannian manifold is a pair ( , g) consisting of smooth 푈 푀 manifold and Riemannian metric g. A manifold may carry a (d) ⋃ is푈 푖푖 a homeomorphism푀 from onto an open subset of 푀 ′ further structure if it is endowed with a metric tensor, which is a 푖 . 푖 푖 휑 푈 푈 natural generation푀 of the inner product between two vectors in 푛 ℝ to an arbitrary manifold. Riemannian metrics, affine (e) Given and such that , the map = connections,푛 parallel transport, curvature tensors, torsion tensors, ( ( ) killingℝ vector fields and conformal killing vector fields play from푖 푗 ) to 푖 푗 is infinitely푖푗 −1 푈 푈 푈 ∩ 푈 ≠ ∅ 휓 important role to develop the theorem of Riemannian manifolds.
    [Show full text]
  • Geodesic Spheres in Grassmann Manifolds
    GEODESIC SPHERES IN GRASSMANN MANIFOLDS BY JOSEPH A. WOLF 1. Introduction Let G,(F) denote the Grassmann manifold consisting of all n-dimensional subspaces of a left /c-dimensional hermitian vectorspce F, where F is the real number field, the complex number field, or the algebra of real quater- nions. We view Cn, (1') tS t Riemnnian symmetric space in the usual way, and study the connected totally geodesic submanifolds B in which any two distinct elements have zero intersection as subspaces of F*. Our main result (Theorem 4 in 8) states that the submanifold B is a compact Riemannian symmetric spce of rank one, and gives the conditions under which it is a sphere. The rest of the paper is devoted to the classification (up to a global isometry of G,(F)) of those submanifolds B which ure isometric to spheres (Theorem 8 in 13). If B is not a sphere, then it is a real, complex, or quater- nionic projective space, or the Cyley projective plane; these submanifolds will be studied in a later paper [11]. The key to this study is the observation thut ny two elements of B, viewed as subspaces of F, are at a constant angle (isoclinic in the sense of Y.-C. Wong [12]). Chapter I is concerned with sets of pairwise isoclinic n-dimen- sional subspces of F, and we are able to extend Wong's structure theorem for such sets [12, Theorem 3.2, p. 25] to the complex numbers nd the qua- ternions, giving a unified and basis-free treatment (Theorem 1 in 4).
    [Show full text]