Differential Geometry II
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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. -
Stable Higgs Bundles on Ruled Surfaces 3
STABLE HIGGS BUNDLES ON RULED SURFACES SNEHAJIT MISRA Abstract. Let π : X = PC(E) −→ C be a ruled surface over an algebraically closed field k of characteristic 0, with a fixed polarization L on X. In this paper, we show that pullback of a (semi)stable Higgs bundle on C under π is a L-(semi)stable Higgs bundle. Conversely, if (V,θ) ∗ is a L-(semi)stable Higgs bundle on X with c1(V ) = π (d) for some divisor d of degree d on C and c2(V ) = 0, then there exists a (semi)stable Higgs bundle (W, ψ) of degree d on C whose pullback under π is isomorphic to (V,θ). As a consequence, we get an isomorphism between the corresponding moduli spaces of (semi)stable Higgs bundles. We also show the existence of non-trivial stable Higgs bundle on X whenever g(C) ≥ 2 and the base field is C. 1. Introduction A Higgs bundle on an algebraic variety X is a pair (V, θ) consisting of a vector bundle V 1 over X together with a Higgs field θ : V −→ V ⊗ ΩX such that θ ∧ θ = 0. Higgs bundle comes with a natural stability condition (see Definition 2.3 for stability), which allows one to study the moduli spaces of stable Higgs bundles on X. Higgs bundles on Riemann surfaces were first introduced by Nigel Hitchin in 1987 and subsequently, Simpson extended this notion on higher dimensional varieties. Since then, these objects have been studied by many authors, but very little is known about stability of Higgs bundles on ruled surfaces. -
Vector Bundles on Projective Space
Vector Bundles on Projective Space Takumi Murayama December 1, 2013 1 Preliminaries on vector bundles Let X be a (quasi-projective) variety over k. We follow [Sha13, Chap. 6, x1.2]. Definition. A family of vector spaces over X is a morphism of varieties π : E ! X −1 such that for each x 2 X, the fiber Ex := π (x) is isomorphic to a vector space r 0 0 Ak(x).A morphism of a family of vector spaces π : E ! X and π : E ! X is a morphism f : E ! E0 such that the following diagram commutes: f E E0 π π0 X 0 and the map fx : Ex ! Ex is linear over k(x). f is an isomorphism if fx is an isomorphism for all x. A vector bundle is a family of vector spaces that is locally trivial, i.e., for each x 2 X, there exists a neighborhood U 3 x such that there is an isomorphism ': π−1(U) !∼ U × Ar that is an isomorphism of families of vector spaces by the following diagram: −1 ∼ r π (U) ' U × A (1.1) π pr1 U −1 where pr1 denotes the first projection. We call π (U) ! U the restriction of the vector bundle π : E ! X onto U, denoted by EjU . r is locally constant, hence is constant on every irreducible component of X. If it is constant everywhere on X, we call r the rank of the vector bundle. 1 The following lemma tells us how local trivializations of a vector bundle glue together on the entire space X. -
LECTURE 6: FIBER BUNDLES in This Section We Will Introduce The
LECTURE 6: FIBER BUNDLES In this section we will introduce the interesting class of fibrations given by fiber bundles. Fiber bundles play an important role in many geometric contexts. For example, the Grassmaniann varieties and certain fiber bundles associated to Stiefel varieties are central in the classification of vector bundles over (nice) spaces. The fact that fiber bundles are examples of Serre fibrations follows from Theorem ?? which states that being a Serre fibration is a local property. 1. Fiber bundles and principal bundles Definition 6.1. A fiber bundle with fiber F is a map p: E ! X with the following property: every ∼ −1 point x 2 X has a neighborhood U ⊆ X for which there is a homeomorphism φU : U × F = p (U) such that the following diagram commutes in which π1 : U × F ! U is the projection on the first factor: φ U × F U / p−1(U) ∼= π1 p * U t Remark 6.2. The projection X × F ! X is an example of a fiber bundle: it is called the trivial bundle over X with fiber F . By definition, a fiber bundle is a map which is `locally' homeomorphic to a trivial bundle. The homeomorphism φU in the definition is a local trivialization of the bundle, or a trivialization over U. Let us begin with an interesting subclass. A fiber bundle whose fiber F is a discrete space is (by definition) a covering projection (with fiber F ). For example, the exponential map R ! S1 is a covering projection with fiber Z. Suppose X is a space which is path-connected and locally simply connected (in fact, the weaker condition of being semi-locally simply connected would be enough for the following construction). -
Functor and Natural Operators on Symplectic Manifolds
Mathematical Methods and Techniques in Engineering and Environmental Science Functor and natural operators on symplectic manifolds CONSTANTIN PĂTRĂŞCOIU Faculty of Engineering and Management of Technological Systems, Drobeta Turnu Severin University of Craiova Drobeta Turnu Severin, Str. Călugăreni, No.1 ROMANIA [email protected] http://www.imst.ro Abstract. Symplectic manifolds arise naturally in abstract formulations of classical Mechanics because the phase–spaces in Hamiltonian mechanics is the cotangent bundle of configuration manifolds equipped with a symplectic structure. The natural functors and operators described in this paper can be helpful both for a unified description of specific proprieties of symplectic manifolds and for finding links to various fields of geometric objects with applications in Hamiltonian mechanics. Key-words: Functors, Natural operators, Symplectic manifolds, Complex structure, Hamiltonian. 1. Introduction. diffeomorphism f : M → N , a vector bundle morphism The geometrics objects (like as vectors, covectors, T*f : T*M→ T*N , which covers f , where −1 tensors, metrics, e.t.c.) on a smooth manifold M, x = x :)*)(()*( x * → xf )( * NTMTfTfT are the elements of the total spaces of a vector So, the cotangent functor, )(:* → VBmManT and bundles with base M. The fields of such geometrics objects are the section in corresponding vector more general ΛkT * : Man(m) → VB , are the bundles. bundle functors from Man(m) to VB, where Man(m) By example, the vectors on a smooth manifold M (subcategory of Man) is the category of m- are the elements of total space of vector bundles dimensional smooth manifolds, the morphisms of this (TM ,π M , M); the covectors on M are the elements category are local diffeomorphisms between of total space of vector bundles (T*M , pM , M). -
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. -
FOLIATIONS Introduction. the Study of Foliations on Manifolds Has a Long
BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY Volume 80, Number 3, May 1974 FOLIATIONS BY H. BLAINE LAWSON, JR.1 TABLE OF CONTENTS 1. Definitions and general examples. 2. Foliations of dimension-one. 3. Higher dimensional foliations; integrability criteria. 4. Foliations of codimension-one; existence theorems. 5. Notions of equivalence; foliated cobordism groups. 6. The general theory; classifying spaces and characteristic classes for foliations. 7. Results on open manifolds; the classification theory of Gromov-Haefliger-Phillips. 8. Results on closed manifolds; questions of compact leaves and stability. Introduction. The study of foliations on manifolds has a long history in mathematics, even though it did not emerge as a distinct field until the appearance in the 1940's of the work of Ehresmann and Reeb. Since that time, the subject has enjoyed a rapid development, and, at the moment, it is the focus of a great deal of research activity. The purpose of this article is to provide an introduction to the subject and present a picture of the field as it is currently evolving. The treatment will by no means be exhaustive. My original objective was merely to summarize some recent developments in the specialized study of codimension-one foliations on compact manifolds. However, somewhere in the writing I succumbed to the temptation to continue on to interesting, related topics. The end product is essentially a general survey of new results in the field with, of course, the customary bias for areas of personal interest to the author. Since such articles are not written for the specialist, I have spent some time in introducing and motivating the subject. -
Dirac Operators for Matrix Algebras Converging to Coadjoint Orbits
DIRAC OPERATORS FOR MATRIX ALGEBRAS CONVERGING TO COADJOINT ORBITS MARC A. RIEFFEL Abstract. In the high-energy physics literature one finds statements such as “matrix algebras converge to the sphere”. Earlier I provided a general precise setting for understanding such statements, in which the matrix algebras are viewed as quantum metric spaces, and convergence is with respect to a quantum Gromov-Hausdorff-type distance. But physicists want even more to treat structures on spheres (and other spaces), such as vector bundles, Yang-Mills functionals, Dirac operators, etc., and they want to approximate these by corresponding structures on matrix algebras. In the present paper we provide a somewhat unified construction of Dirac operators on coadjoint orbits and on the matrix algebras that converge to them. This enables us to prove our main theorem, whose content is that, for the quantum metric-space structures determined by the Dirac operators that we construct, the matrix algebras do indeed converge to the coadjoint orbits, for a quite strong version of quantum Gromov-Hausdorff distance. Contents Introduction 2 1. Ergodic actions and Dirac operators 5 2. Invariance under the group actions 9 3. Charge Conjugation 11 4. Casimir operators 12 5. First facts about spinors 14 6. The C*-metrics 16 7. More about spinors 20 8. Dirac operators for matrix algebras 21 arXiv:2108.01136v1 [math.OA] 2 Aug 2021 9. The fuzzy sphere 23 10. Homogeneous spaces and the Dirac operator for G 29 11. Dirac operators for homogeneous spaces 30 12. Complex structure on coadjoint orbits 36 13. Yet more about spinors 40 14. -
Fluxes, Bundle Gerbes and 2-Hilbert Spaces
Lett Math Phys DOI 10.1007/s11005-017-0971-x Fluxes, bundle gerbes and 2-Hilbert spaces Severin Bunk1 · Richard J. Szabo2 Received: 12 January 2017 / Revised: 13 April 2017 / Accepted: 8 May 2017 © The Author(s) 2017. This article is an open access publication Abstract We elaborate on the construction of a prequantum 2-Hilbert space from a bundle gerbe over a 2-plectic manifold, providing the first steps in a programme of higher geometric quantisation of closed strings in flux compactifications and of M5- branes in C-fields. We review in detail the construction of the 2-category of bundle gerbes and introduce the higher geometrical structures necessary to turn their cate- gories of sections into 2-Hilbert spaces. We work out several explicit examples of 2-Hilbert spaces in the context of closed strings and M5-branes on flat space. We also work out the prequantum 2-Hilbert space associated with an M-theory lift of closed strings described by an asymmetric cyclic orbifold of the SU(2) WZW model, pro- viding an example of sections of a torsion gerbe on a curved background. We describe the dimensional reduction of M-theory to string theory in these settings as a map from 2-isomorphism classes of sections of bundle gerbes to sections of corresponding line bundles, which is compatible with the respective monoidal structures and module actions. Keywords Bundle gerbes · Higher geometric quantisation · Fluxes in string and M-theory · Higher geometric structures Mathematics Subject Classification 81S10 · 53Z05 · 18F99 B Severin Bunk [email protected] Richard J. -
Flat Vector Bundles and Analytic Torsion on Orbifolds
i “Orbifold_Torsion_final” — 2020/9/10 — 0:54 — page 1 — #1 i i i Communications in Analysis and Geometry Volume 00, Number 0, 1–85, 0000 Flat vector bundles and analytic torsion on orbifolds Shu Shen and Jianqing Yu This article is devoted to a study of flat orbifold vector bundles. We construct a bijection between the isomorphic classes of proper flat orbifold vector bundles and the equivalence classes of represen- tations of the orbifold fundamental groups of base orbifolds. We establish a Bismut-Zhang like anomaly formula for the Ray-Singer metric on the determinant line of the cohomology of a compact orb- ifold with coefficients in an orbifold flat vector bundle. We show that the analytic torsion of an acyclic unitary flat orbifold vector bundle is equal to the value at zero of a dynamical zeta function when the underlying orbifold is a compact locally symmetric space of reductive type, which extends one of the results obtained by the first author for compact locally symmetric manifolds. Introduction 3 1 Preliminary 10 1.1 Determinants 11 1.2 Group actions 11 1.3 Characteristic forms on manifolds 11 2 Topology of orbifolds 15 arXiv:1704.08369v3 [math.DG] 9 Sep 2020 2.1 Definition of orbifolds 15 2.2 Singular set of orbifolds 21 2.3 Orbifold vector bundle 22 1 i i i i i “Orbifold_Torsion_final” — 2020/9/10 — 0:54 — page 2 — #2 i i i 2 S. Shen and J. Yu 2.4 Orbifold fundamental groups and universal covering orbifold 26 2.5 Flat vector bundles and holonomy 32 3 Differential calculus on orbifolds 35 3.1 Differential operators on orbifolds 36 3.2