Gravity and Connections on Vector Bundles
Total Page:16
File Type:pdf, Size:1020Kb
Load more
Recommended publications
-
Equivalence of A-Approximate Continuity for Self-Adjoint
Equivalence of A-Approximate Continuity for Self-Adjoint Expansive Linear Maps a,1 b, ,2 Sz. Gy. R´ev´esz , A. San Antol´ın ∗ aA. R´enyi Institute of Mathematics, Hungarian Academy of Sciences, Budapest, P.O.B. 127, 1364 Hungary bDepartamento de Matem´aticas, Universidad Aut´onoma de Madrid, 28049 Madrid, Spain Abstract Let A : Rd Rd, d 1, be an expansive linear map. The notion of A-approximate −→ ≥ continuity was recently used to give a characterization of scaling functions in a multiresolution analysis (MRA). The definition of A-approximate continuity at a point x – or, equivalently, the definition of the family of sets having x as point of A-density – depend on the expansive linear map A. The aim of the present paper is to characterize those self-adjoint expansive linear maps A , A : Rd Rd for which 1 2 → the respective concepts of Aµ-approximate continuity (µ = 1, 2) coincide. These we apply to analyze the equivalence among dilation matrices for a construction of systems of MRA. In particular, we give a full description for the equivalence class of the dyadic dilation matrix among all self-adjoint expansive maps. If the so-called “four exponentials conjecture” of algebraic number theory holds true, then a similar full description follows even for general self-adjoint expansive linear maps, too. arXiv:math/0703349v2 [math.CA] 7 Sep 2007 Key words: A-approximate continuity, multiresolution analysis, point of A-density, self-adjoint expansive linear map. 1 Supported in part in the framework of the Hungarian-Spanish Scientific and Technological Governmental Cooperation, Project # E-38/04. -
Arxiv:2009.07259V1 [Math.AP] 15 Sep 2020
A GEOMETRIC TRAPPING APPROACH TO GLOBAL REGULARITY FOR 2D NAVIER-STOKES ON MANIFOLDS AYNUR BULUT AND KHANG MANH HUYNH Abstract. In this paper, we use frequency decomposition techniques to give a direct proof of global existence and regularity for the Navier-Stokes equations on two-dimensional Riemannian manifolds without boundary. Our techniques are inspired by an approach of Mattingly and Sinai [15] which was developed in the context of periodic boundary conditions on a flat background, and which is based on a maximum principle for Fourier coefficients. The extension to general manifolds requires several new ideas, connected to the less favorable spectral localization properties in our setting. Our argu- ments make use of frequency projection operators, multilinear estimates that originated in the study of the non-linear Schr¨odingerequation, and ideas from microlocal analysis. 1. Introduction Let (M; g) be a closed, oriented, connected, compact smooth two-dimensional Riemannian manifold, and let X(M) denote the space of smooth vector fields on M. We consider the incompressible Navier-Stokes equations on M, with viscosity coefficient ν > 0, @ U + div (U ⊗ U) − ν∆ U = − grad p in M t M ; (1) div U = 0 in M with initial data U0 2 X(M); where I ⊂ R is an open interval, and where U : I ! X(M) and p : I × M ! R represent the velocity and pressure of the fluid, respectively. Here, the operator ∆M is any choice of Laplacian defined on vector fields on M, discussed below. arXiv:2009.07259v1 [math.AP] 15 Sep 2020 The theory of two-dimensional fluid flows on flat spaces is well-developed, and a variety of global regularity results are well-known. -
Introduction to Gauge Theory Arxiv:1910.10436V1 [Math.DG] 23
Introduction to Gauge Theory Andriy Haydys 23rd October 2019 This is lecture notes for a course given at the PCMI Summer School “Quantum Field The- ory and Manifold Invariants” (July 1 – July 5, 2019). I describe basics of gauge-theoretic approach to construction of invariants of manifolds. The main example considered here is the Seiberg–Witten gauge theory. However, I tried to present the material in a form, which is suitable for other gauge-theoretic invariants too. Contents 1 Introduction2 2 Bundles and connections4 2.1 Vector bundles . .4 2.1.1 Basic notions . .4 2.1.2 Operations on vector bundles . .5 2.1.3 Sections . .6 2.1.4 Covariant derivatives . .6 2.1.5 The curvature . .8 2.1.6 The gauge group . 10 2.2 Principal bundles . 11 2.2.1 The frame bundle and the structure group . 11 2.2.2 The associated vector bundle . 14 2.2.3 Connections on principal bundles . 16 2.2.4 The curvature of a connection on a principal bundle . 19 arXiv:1910.10436v1 [math.DG] 23 Oct 2019 2.2.5 The gauge group . 21 2.3 The Levi–Civita connection . 22 2.4 Classification of U(1) and SU(2) bundles . 23 2.4.1 Complex line bundles . 24 2.4.2 Quaternionic line bundles . 25 3 The Chern–Weil theory 26 3.1 The Chern–Weil theory . 26 3.1.1 The Chern classes . 28 3.2 The Chern–Simons functional . 30 3.3 The modui space of flat connections . 32 3.3.1 Parallel transport and holonomy . -
LECTURES on PURE SPINORS and MOMENT MAPS Contents 1
LECTURES ON PURE SPINORS AND MOMENT MAPS E. MEINRENKEN Contents 1. Introduction 1 2. Volume forms on conjugacy classes 1 3. Clifford algebras and spinors 3 4. Linear Dirac geometry 8 5. The Cartan-Dirac structure 11 6. Dirac structures 13 7. Group-valued moment maps 16 References 20 1. Introduction This article is an expanded version of notes for my lectures at the summer school on `Poisson geometry in mathematics and physics' at Keio University, Yokohama, June 5{9 2006. The plan of these lectures was to give an elementary introduction to the theory of Dirac structures, with applications to Lie group valued moment maps. Special emphasis was given to the pure spinor approach to Dirac structures, developed in Alekseev-Xu [7] and Gualtieri [20]. (See [11, 12, 16] for the more standard approach.) The connection to moment maps was made in the work of Bursztyn-Crainic [10]. Parts of these lecture notes are based on a forthcoming joint paper [1] with Anton Alekseev and Henrique Bursztyn. I would like to thank the organizers of the school, Yoshi Maeda and Guiseppe Dito, for the opportunity to deliver these lectures, and for a greatly enjoyable meeting. I also thank Yvette Kosmann-Schwarzbach and the referee for a number of helpful comments. 2. Volume forms on conjugacy classes We will begin with the following FACT, which at first sight may seem quite unrelated to the theme of these lectures: FACT. Let G be a simply connected semi-simple real Lie group. Then every conjugacy class in G carries a canonical invariant volume form. -
Lecture Notes in Mathematics
Lecture Notes in Mathematics Edited by A. Dold and B. Eckmann Subseries: Mathematisches Institut der Universit~it und Max-Planck-lnstitut for Mathematik, Bonn - voL 5 Adviser: E Hirzebruch 1111 Arbeitstagung Bonn 1984 Proceedings of the meeting held by the Max-Planck-lnstitut fur Mathematik, Bonn June 15-22, 1984 Edited by E Hirzebruch, J. Schwermer and S. Suter I IIII Springer-Verlag Berlin Heidelberg New York Tokyo Herausgeber Friedrich Hirzebruch Joachim Schwermer Silke Suter Max-Planck-lnstitut fLir Mathematik Gottfried-Claren-Str. 26 5300 Bonn 3, Federal Republic of Germany AMS-Subject Classification (1980): 10D15, 10D21, 10F99, 12D30, 14H10, 14H40, 14K22, 17B65, 20G35, 22E47, 22E65, 32G15, 53C20, 57 N13, 58F19 ISBN 3-54045195-8 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15195-8 Springer-Verlag New York Heidelberg Berlin Tokyo CIP-Kurztitelaufnahme der Deutschen Bibliothek. Mathematische Arbeitstagung <25. 1984. Bonn>: Arbeitstagung Bonn: 1984; proceedings of the meeting, held in Bonn, June 15-22, 1984 / [25. Math. Arbeitstagung]. Ed. by E Hirzebruch ... - Berlin; Heidelberg; NewYork; Tokyo: Springer, 1985. (Lecture notes in mathematics; Vol. 1t11: Subseries: Mathematisches I nstitut der U niversit~it und Max-Planck-lnstitut for Mathematik Bonn; VoL 5) ISBN 3-540-t5195-8 (Berlin...) ISBN 0-387q5195-8 (NewYork ...) NE: Hirzebruch, Friedrich [Hrsg.]; Lecture notes in mathematics / Subseries: Mathematischee Institut der UniversitAt und Max-Planck-lnstitut fur Mathematik Bonn; HST This work ts subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re~use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. -
NOTES on DIFFERENTIAL FORMS. PART 3: TENSORS 1. What Is A
NOTES ON DIFFERENTIAL FORMS. PART 3: TENSORS 1. What is a tensor? 1 n Let V be a finite-dimensional vector space. It could be R , it could be the tangent space to a manifold at a point, or it could just be an abstract vector space. A k-tensor is a map T : V × · · · × V ! R 2 (where there are k factors of V ) that is linear in each factor. That is, for fixed ~v2; : : : ;~vk, T (~v1;~v2; : : : ;~vk−1;~vk) is a linear function of ~v1, and for fixed ~v1;~v3; : : : ;~vk, T (~v1; : : : ;~vk) is a k ∗ linear function of ~v2, and so on. The space of k-tensors on V is denoted T (V ). Examples: n • If V = R , then the inner product P (~v; ~w) = ~v · ~w is a 2-tensor. For fixed ~v it's linear in ~w, and for fixed ~w it's linear in ~v. n • If V = R , D(~v1; : : : ;~vn) = det ~v1 ··· ~vn is an n-tensor. n • If V = R , T hree(~v) = \the 3rd entry of ~v" is a 1-tensor. • A 0-tensor is just a number. It requires no inputs at all to generate an output. Note that the definition of tensor says nothing about how things behave when you rotate vectors or permute their order. The inner product P stays the same when you swap the two vectors, but the determinant D changes sign when you swap two vectors. Both are tensors. For a 1-tensor like T hree, permuting the order of entries doesn't even make sense! ~ ~ Let fb1;:::; bng be a basis for V . -
Tensor Products
Tensor products Joel Kamnitzer April 5, 2011 1 The definition Let V, W, X be three vector spaces. A bilinear map from V × W to X is a function H : V × W → X such that H(av1 + v2, w) = aH(v1, w) + H(v2, w) for v1,v2 ∈ V, w ∈ W, a ∈ F H(v, aw1 + w2) = aH(v, w1) + H(v, w2) for v ∈ V, w1, w2 ∈ W, a ∈ F Let V and W be vector spaces. A tensor product of V and W is a vector space V ⊗ W along with a bilinear map φ : V × W → V ⊗ W , such that for every vector space X and every bilinear map H : V × W → X, there exists a unique linear map T : V ⊗ W → X such that H = T ◦ φ. In other words, giving a linear map from V ⊗ W to X is the same thing as giving a bilinear map from V × W to X. If V ⊗ W is a tensor product, then we write v ⊗ w := φ(v ⊗ w). Note that there are two pieces of data in a tensor product: a vector space V ⊗ W and a bilinear map φ : V × W → V ⊗ W . Here are the main results about tensor products summarized in one theorem. Theorem 1.1. (i) Any two tensor products of V, W are isomorphic. (ii) V, W has a tensor product. (iii) If v1,...,vn is a basis for V and w1,...,wm is a basis for W , then {vi ⊗ wj }1≤i≤n,1≤j≤m is a basis for V ⊗ W . -
Determinants in Geometric Algebra
Determinants in Geometric Algebra Eckhard Hitzer 16 June 2003, recovered+expanded May 2020 1 Definition Let f be a linear map1, of a real linear vector space Rn into itself, an endomor- phism n 0 n f : a 2 R ! a 2 R : (1) This map is extended by outermorphism (symbol f) to act linearly on multi- vectors f(a1 ^ a2 ::: ^ ak) = f(a1) ^ f(a2) ::: ^ f(ak); k ≤ n: (2) By definition f is grade-preserving and linear, mapping multivectors to mul- tivectors. Examples are the reflections, rotations and translations described earlier. The outermorphism of a product of two linear maps fg is the product of the outermorphisms f g f[g(a1)] ^ f[g(a2)] ::: ^ f[g(ak)] = f[g(a1) ^ g(a2) ::: ^ g(ak)] = f[g(a1 ^ a2 ::: ^ ak)]; (3) with k ≤ n. The square brackets can safely be omitted. The n{grade pseudoscalars of a geometric algebra are unique up to a scalar factor. This can be used to define the determinant2 of a linear map as det(f) = f(I)I−1 = f(I) ∗ I−1; and therefore f(I) = det(f)I: (4) For an orthonormal basis fe1; e2;:::; eng the unit pseudoscalar is I = e1e2 ::: en −1 q q n(n−1)=2 with inverse I = (−1) enen−1 ::: e1 = (−1) (−1) I, where q gives the number of basis vectors, that square to −1 (the linear space is then Rp;q). According to Grassmann n-grade vectors represent oriented volume elements of dimension n. The determinant therefore shows how these volumes change under linear maps. -
28. Exterior Powers
28. Exterior powers 28.1 Desiderata 28.2 Definitions, uniqueness, existence 28.3 Some elementary facts 28.4 Exterior powers Vif of maps 28.5 Exterior powers of free modules 28.6 Determinants revisited 28.7 Minors of matrices 28.8 Uniqueness in the structure theorem 28.9 Cartan's lemma 28.10 Cayley-Hamilton Theorem 28.11 Worked examples While many of the arguments here have analogues for tensor products, it is worthwhile to repeat these arguments with the relevant variations, both for practice, and to be sensitive to the differences. 1. Desiderata Again, we review missing items in our development of linear algebra. We are missing a development of determinants of matrices whose entries may be in commutative rings, rather than fields. We would like an intrinsic definition of determinants of endomorphisms, rather than one that depends upon a choice of coordinates, even if we eventually prove that the determinant is independent of the coordinates. We anticipate that Artin's axiomatization of determinants of matrices should be mirrored in much of what we do here. We want a direct and natural proof of the Cayley-Hamilton theorem. Linear algebra over fields is insufficient, since the introduction of the indeterminate x in the definition of the characteristic polynomial takes us outside the class of vector spaces over fields. We want to give a conceptual proof for the uniqueness part of the structure theorem for finitely-generated modules over principal ideal domains. Multi-linear algebra over fields is surely insufficient for this. 417 418 Exterior powers 2. Definitions, uniqueness, existence Let R be a commutative ring with 1. -
Math 395. Tensor Products and Bases Let V and V Be Finite-Dimensional
Math 395. Tensor products and bases Let V and V 0 be finite-dimensional vector spaces over a field F . Recall that a tensor product of V and V 0 is a pait (T, t) consisting of a vector space T over F and a bilinear pairing t : V × V 0 → T with the following universal property: for any bilinear pairing B : V × V 0 → W to any vector space W over F , there exists a unique linear map L : T → W such that B = L ◦ t. Roughly speaking, t “uniquely linearizes” all bilinear pairings of V and V 0 into arbitrary F -vector spaces. In class it was proved that if (T, t) and (T 0, t0) are two tensor products of V and V 0, then there exists a unique linear isomorphism T ' T 0 carrying t and t0 (and vice-versa). In this sense, the tensor product of V and V 0 (equipped with its “universal” bilinear pairing from V × V 0!) is unique up to unique isomorphism, and so we may speak of “the” tensor product of V and V 0. You must never forget to think about the data of t when you contemplate the tensor product of V and V 0: it is the pair (T, t) and not merely the underlying vector space T that is the focus of interest. In this handout, we review a method of construction of tensor products (there is another method that involved no choices, but is horribly “big”-looking and is needed when considering modules over commutative rings) and we work out some examples related to the construction. -
The Divergence Theorem Cartan's Formula II. for Any Smooth Vector
The Divergence Theorem Cartan’s Formula II. For any smooth vector field X and any smooth differential form ω, LX = iX d + diX . Lemma. Let x : U → Rn be a positively oriented chart on (M,G), with volume j ∂ form vM , and X = Pj X ∂xj . Then, we have U √ 1 n ∂( gXj) L v = di v = √ X , X M X M g ∂xj j=1 and √ 1 n ∂( gXj ) tr DX = √ X . g ∂xj j=1 Proof. (i) We have √ 1 n LX vM =diX vM = d(iX gdx ∧···∧dx ) n √ =dX(−1)j−1 gXjdx1 ∧···∧dxj ∧···∧dxn d j=1 n √ = X(−1)j−1d( gXj) ∧ dx1 ∧···∧dxj ∧···∧dxn d j=1 √ n ∂( gXj ) = X dx1 ∧···∧dxn ∂xj j=1 √ 1 n ∂( gXj ) =√ X v . g ∂xj M j=1 ∂X` ` k ∂ (ii) We have D∂/∂xj X = P` ∂xj + Pk Γkj X ∂x` , which implies ∂X` tr DX = X + X Γ` Xk. ∂x` k` ` k Since 1 Γ` = X g`r{∂ g + ∂ g − ∂ g } k` 2 k `r ` kr r k` r,` 1 == X g`r∂ g 2 k `r r,` √ 1 ∂ g ∂ g = k = √k , 2 g g √ √ ∂X` X` ∂ g 1 n ∂( gXj ) tr DX = X + √ k } = √ X . ∂x` g ∂x` g ∂xj ` j=1 Typeset by AMS-TEX 1 2 Corollary. Let (M,g) be an oriented Riemannian manifold. Then, for any X ∈ Γ(TM), d(iX dvg)=tr DX =(div X)dvg. Stokes’ Theorem. Let M be a smooth, oriented n-dimensional manifold with boundary. Let ω be a compactly supported smooth (n − 1)-form on M. -
Differential Forms Diff Geom II, WS 2015/16
J.M. Sullivan, TU Berlin B: Differential Forms Diff Geom II, WS 2015/16 B. DIFFERENTIAL FORMS instance, if S has k elements this gives a k-dimensional vector space with S as basis. We have already seen one-forms (covector fields) on a Given vector spaces V and W, let F be the free vector space over the set V × W. (This consists of formal sums manifold. In general, a k-form is a field of alternating k- P linear forms on the tangent spaces of a manifold. Forms ai(vi, wi) but ignores all the structure we have on the set are the natural objects for integration: a k-form can be in- V × W.) Now let R ⊂ F be the linear subspace spanned by tegrated over an oriented k-submanifold. We start with ten- all elements of the form: sor products and the exterior algebra of multivectors. (v + v0, w) − (v, w) − (v0, w), (v, w + w0) − (v, w) − (v, w0), (av, w) − a(v, w), (v, aw) − a(v, w). B1. Tensor products These correspond of course to the bilinearity conditions Recall that, if V, W and X are vector spaces, then a map we started with. The quotient vector space F/R will be the b: V × W → X is called bilinear if tensor product V ⊗ W. We have started with all possible v ⊗ w as generators and thrown in just enough relations to b(v + v0, w) = b(v, w) + b(v0, w), make the map (v, w) 7→ v ⊗ w be bilinear. b(v, w + w0) = b(v, w) + b(v, w0), The tensor product is commutative: there is a natural linear isomorphism V⊗W → W⊗V such that v⊗w 7→ w⊗v.