Hodge Theory of SKT Manifolds
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When Is a Connection a Metric Connection?
NEW ZEALAND JOURNAL OF MATHEMATICS Volume 38 (2008), 225–238 WHEN IS A CONNECTION A METRIC CONNECTION? Richard Atkins (Received April 2008) Abstract. There are two sides to this investigation: first, the determination of the integrability conditions that ensure the existence of a local parallel metric in the neighbourhood of a given point and second, the characterization of the topological obstruction to a globally defined parallel metric. The local problem has been previously solved by means of constructing a derived flag. Herein we continue the inquiry to the global aspect, which is formulated in terms of the cohomology of the constant sheaf of sections in the general linear group. 1. Introduction A connection on a manifold is a type of differentiation that acts on vector fields, differential forms and tensor products of these objects. Its importance lies in the fact that given a piecewise continuous curve connecting two points on the manifold, the connection defines a linear isomorphism between the respective tangent spaces at these points. Another fundamental concept in the study of differential geometry is that of a metric, which provides a measure of distance on the manifold. It is well known that a metric uniquely determines a Levi-Civita connection: a symmetric connection for which the metric is parallel. Not all connections are derived from a metric and so it is natural to ask, for a symmetric connection, if there exists a parallel metric, that is, whether the connection is a Levi-Civita connection. More generally, a connection on a manifold M, symmetric or not, is said to be metric if there exists a parallel metric defined on M. -
Lecture 24/25: Riemannian Metrics
Lecture 24/25: Riemannian metrics An inner product on Rn allows us to do the following: Given two curves intersecting at a point x Rn,onecandeterminetheanglebetweentheir tangents: ∈ cos θ = u, v / u v . | || | On a general manifold, we would again like to have an inner product on each TpM.Theniftwocurvesintersectatapointp,onecanmeasuretheangle between them. Of course, there are other geometric invariants that arise out of this structure. 1. Riemannian and Hermitian metrics Definition 24.2. A Riemannian metric on a vector bundle E is a section g Γ(Hom(E E,R)) ∈ ⊗ such that g is symmetric and positive definite. A Riemannian manifold is a manifold together with a choice of Riemannian metric on its tangent bundle. Chit-chat 24.3. A section of Hom(E E,R)isasmoothchoiceofalinear map ⊗ gp : Ep Ep R ⊗ → at every point p.Thatg is symmetric means that for every u, v E ,wehave ∈ p gp(u, v)=gp(v, u). That g is positive definite means that g (v, v) 0 p ≥ for all v E ,andequalityholdsifandonlyifv =0 E . ∈ p ∈ p Chit-chat 24.4. As usual, one can try to understand g in local coordinates. If one chooses a trivializing set of linearly independent sections si ,oneob- tains a matrix of functions { } gij = gp((si)p, (sj)p). By symmetry of g,thisisasymmetricmatrix. 85 n Example 24.5. T R is trivial. Let gij = δij be the constant matrix of functions, so that gij(p)=I is the identity matrix for every point. Then on n n every fiber, g defines the usual inner product on TpR ∼= R . -
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. -
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,. -
Analyzing Non-Degenerate 2-Forms with Riemannian Metrics
EXPOSITIONES Expo. Math. 20 (2002): 329-343 © Urban & Fischer Verlag MATHEMATICAE www.u rba nfischer.de/journals/expomath Analyzing Non-degenerate 2-Forms with Riemannian Metrics Philippe Delano~ Universit~ de Nice-Sophia Antipolis Math~matiques, Parc Valrose, F-06108 Nice Cedex 2, France Abstract. We give a variational proof of the harmonicity of a symplectic form with respect to adapted riemannian metrics, show that a non-degenerate 2-form must be K~ihlerwhenever parallel for some riemannian metric, regardless of adaptation, and discuss k/ihlerness under curvature assmnptions. Introduction In the first part of this paper, we aim at a better understanding of the following folklore result (see [14, pp. 140-141] and references therein): Theorem 1 . Let a be a non-degenerate 2-form on a manifold M and g, a riemannian metric adapted to it. If a is closed, it is co-closed for g. Let us specify at once a bit of terminology. We say a riemannian metric g is adapted to a non-degenerate 2-form a on a 2m-manifold M, if at each point Xo E M there exists a chart of M in which g(Xo) and a(Xo) read like the standard euclidean metric and symplectic form of R 2m. If, moreover, this occurs for the first jets of a (resp. a and g) and the corresponding standard structures of R 2m, the couple (~r,g) is called an almost-K~ihler (resp. a K/ihler) structure; in particular, a is then closed. Given a non-degenerate, an adapted metric g can be constructed out of any riemannian metric on M, using Cartan's polar decomposition (cf. -
Parallel Spinors and Connections with Skew-Symmetric Torsion in String Theory*
ASIAN J. MATH. © 2002 International Press Vol. 6, No. 2, pp. 303-336, June 2002 005 PARALLEL SPINORS AND CONNECTIONS WITH SKEW-SYMMETRIC TORSION IN STRING THEORY* THOMAS FRIEDRICHt AND STEFAN IVANOV* Abstract. We describe all almost contact metric, almost hermitian and G2-structures admitting a connection with totally skew-symmetric torsion tensor, and prove that there exists at most one such connection. We investigate its torsion form, its Ricci tensor, the Dirac operator and the V- parallel spinors. In particular, we obtain partial solutions of the type // string equations in dimension n = 5, 6 and 7. 1. Introduction. Linear connections preserving a Riemannian metric with totally skew-symmetric torsion recently became a subject of interest in theoretical and mathematical physics. For example, the target space of supersymmetric sigma models with Wess-Zumino term carries a geometry of a metric connection with skew-symmetric torsion [23, 34, 35] (see also [42] and references therein). In supergravity theories, the geometry of the moduli space of a class of black holes is carried out by a metric connection with skew-symmetric torsion [27]. The geometry of NS-5 brane solutions of type II supergravity theories is generated by a metric connection with skew-symmetric torsion [44, 45, 43]. The existence of parallel spinors with respect to a metric connection with skew-symmetric torsion on a Riemannian spin manifold is of importance in string theory, since they are associated with some string solitons (BPS solitons) [43]. Supergravity solutions that preserve some of the supersymmetry of the underlying theory have found many applications in the exploration of perturbative and non-perturbative properties of string theory. -
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. -
Tensor Calculus and Differential Geometry
Course Notes Tensor Calculus and Differential Geometry 2WAH0 Luc Florack March 10, 2021 Cover illustration: papyrus fragment from Euclid’s Elements of Geometry, Book II [8]. Contents Preface iii Notation 1 1 Prerequisites from Linear Algebra 3 2 Tensor Calculus 7 2.1 Vector Spaces and Bases . .7 2.2 Dual Vector Spaces and Dual Bases . .8 2.3 The Kronecker Tensor . 10 2.4 Inner Products . 11 2.5 Reciprocal Bases . 14 2.6 Bases, Dual Bases, Reciprocal Bases: Mutual Relations . 16 2.7 Examples of Vectors and Covectors . 17 2.8 Tensors . 18 2.8.1 Tensors in all Generality . 18 2.8.2 Tensors Subject to Symmetries . 22 2.8.3 Symmetry and Antisymmetry Preserving Product Operators . 24 2.8.4 Vector Spaces with an Oriented Volume . 31 2.8.5 Tensors on an Inner Product Space . 34 2.8.6 Tensor Transformations . 36 2.8.6.1 “Absolute Tensors” . 37 CONTENTS i 2.8.6.2 “Relative Tensors” . 38 2.8.6.3 “Pseudo Tensors” . 41 2.8.7 Contractions . 43 2.9 The Hodge Star Operator . 43 3 Differential Geometry 47 3.1 Euclidean Space: Cartesian and Curvilinear Coordinates . 47 3.2 Differentiable Manifolds . 48 3.3 Tangent Vectors . 49 3.4 Tangent and Cotangent Bundle . 50 3.5 Exterior Derivative . 51 3.6 Affine Connection . 52 3.7 Lie Derivative . 55 3.8 Torsion . 55 3.9 Levi-Civita Connection . 56 3.10 Geodesics . 57 3.11 Curvature . 58 3.12 Push-Forward and Pull-Back . 59 3.13 Examples . 60 3.13.1 Polar Coordinates in the Euclidean Plane . -
The Language of Differential Forms
Appendix A The Language of Differential Forms This appendix—with the only exception of Sect.A.4.2—does not contain any new physical notions with respect to the previous chapters, but has the purpose of deriving and rewriting some of the previous results using a different language: the language of the so-called differential (or exterior) forms. Thanks to this language we can rewrite all equations in a more compact form, where all tensor indices referred to the diffeomorphisms of the curved space–time are “hidden” inside the variables, with great formal simplifications and benefits (especially in the context of the variational computations). The matter of this appendix is not intended to provide a complete nor a rigorous introduction to this formalism: it should be regarded only as a first, intuitive and oper- ational approach to the calculus of differential forms (also called exterior calculus, or “Cartan calculus”). The main purpose is to quickly put the reader in the position of understanding, and also independently performing, various computations typical of a geometric model of gravity. The readers interested in a more rigorous discussion of differential forms are referred, for instance, to the book [22] of the bibliography. Let us finally notice that in this appendix we will follow the conventions introduced in Chap. 12, Sect. 12.1: latin letters a, b, c,...will denote Lorentz indices in the flat tangent space, Greek letters μ, ν, α,... tensor indices in the curved manifold. For the matter fields we will always use natural units = c = 1. Also, unless otherwise stated, in the first three Sects. -
Operator-Valued Tensors on Manifolds: a Framework for Field
Operator-Valued Tensors on Manifolds: A Framework for Field Quantization 1 2 H. Feizabadi , N. Boroojerdian ∗ 1,2Department of pure Mathematics, Faculty of Mathematics and Computer Science, Amirkabir University of Technology, No. 424, Hafez Ave., Tehran, Iran. Abstract In this paper we try to prepare a framework for field quantization. To this end, we aim to replace the field of scalars R by self-adjoint elements of a commu- tative C⋆-algebra, and reach an appropriate generalization of geometrical concepts on manifolds. First, we put forward the concept of operator-valued tensors and extend semi-Riemannian metrics to operator valued metrics. Then, in this new geometry, some essential concepts of Riemannian geometry such as curvature ten- sor, Levi-Civita connection, Hodge star operator, exterior derivative, divergence,... will be considered. Keywords: Operator-valued tensors, Operator-Valued Semi-Riemannian Met- rics, Levi-Civita Connection, Curvature, Hodge star operator MSC(2010): Primary: 65F05; Secondary: 46L05, 11Y50. 0 Introduction arXiv:1501.05065v2 [math-ph] 27 Jan 2015 The aim of the present paper is to extend the theory of semi-Riemannian metrics to operator-valued semi-Riemannian metrics, which may be used as a framework for field quantization. Spaces of linear operators are directly related to C∗-algebras, so C∗- algebras are considered as a basic concept in this article. This paper has its roots in Hilbert C⋆-modules, which are frequently used in the theory of operator algebras, allowing one to obtain information about C⋆-algebras by studying Hilbert C⋆-modules over them. Hilbert C⋆-modules provide a natural generalization of Hilbert spaces arising when the field of scalars C is replaced by an arbitrary C⋆-algebra. -
3. Introducing Riemannian Geometry
3. Introducing Riemannian Geometry We have yet to meet the star of the show. There is one object that we can place on a manifold whose importance dwarfs all others, at least when it comes to understanding gravity. This is the metric. The existence of a metric brings a whole host of new concepts to the table which, collectively, are called Riemannian geometry.Infact,strictlyspeakingwewillneeda slightly di↵erent kind of metric for our study of gravity, one which, like the Minkowski metric, has some strange minus signs. This is referred to as Lorentzian Geometry and a slightly better name for this section would be “Introducing Riemannian and Lorentzian Geometry”. However, for our immediate purposes the di↵erences are minor. The novelties of Lorentzian geometry will become more pronounced later in the course when we explore some of the physical consequences such as horizons. 3.1 The Metric In Section 1, we informally introduced the metric as a way to measure distances between points. It does, indeed, provide this service but it is not its initial purpose. Instead, the metric is an inner product on each vector space Tp(M). Definition:Ametric g is a (0, 2) tensor field that is: Symmetric: g(X, Y )=g(Y,X). • Non-Degenerate: If, for any p M, g(X, Y ) =0forallY T (M)thenX =0. • 2 p 2 p p With a choice of coordinates, we can write the metric as g = g (x) dxµ dx⌫ µ⌫ ⌦ The object g is often written as a line element ds2 and this expression is abbreviated as 2 µ ⌫ ds = gµ⌫(x) dx dx This is the form that we saw previously in (1.4). -
Torsion Tensor and Its Geometric Interpretation
Annales de la Fondation Louis de Broglie, Volume 32 no 2-3, 2007 195 Torsion tensor and its geometric interpretation S. Capozziello and C. Stornaiolo Dipartimento di Scienze Fisiche and INFN Sez. di Napoli, Universit`a “Federico II” di Napoli, Via Cintia, Edificio N’ I-80126 Napoli, Italy ABSTRACT. Generally, spin is considered to be the source of torsion, but there are several other possibilities in which torsion emerges in different contexts. In some cases a phenomenological counterpart is absent, in some other cases torsion arises from sources without spin as a gradient of a scalar field. Accordingly, we propose two classifi- cation schemes. The first one is based on the possibility to construct torsion tensors from the product of a covariant bivector and a vector and their respective space-time properties. The second one is obtained by starting from the decomposition of torsion into three irreducible pieces. Their space-time properties again lead to a complete classifica- tion. The classifications found are given in a U4, a four dimensional space-time where the torsion tensors have some peculiar properties. The irreducible decomposition is useful since most of the phenomeno- logical work done for torsion concerns four dimensional cosmological models. In the second part of the paper two applications of these clas- sification schemes are given. The modifications of energy-momentum tensors are considered that arise due to different sources of torsion. Furthermore, we analyze the contributions of torsion to shear, vortic- ity, expansion and acceleration. Finally the generalized Raychaudhuri equation is discussed. Keywords: torsion, U4 space-time, ECSK theory.