DOCSLIB.ORG

Connections on Principal Bundles

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Connections on Principal Bundles Connections on principal bundles Santiago Quintero de los Ríos* December 16, 2020 Contents 1 Connections on principal bundles 1 1.1 Connections as horizontal distributions ........................ 1 1.2 Connections as 1-forms ................................ 3 1.3 Local expressions, or, why physicists did nothing wrong ............... 5 1.4 Horizontal lifts, parallel transport and holonomy ................... 8 2 Curvature 11 2.1 The curvature 2-form and structure equation ..................... 11 2.2 Local expressions (Curvature Edition) ......................... 14 2.3 The exterior covariant derivative ........................... 15 3 The relation with connections on vector bundles 17 3.1 From vector bundles to principal bundles ....................... 17 3.2 Interlude: Associated bundles ............................. 19 3.3 From principal bundles to vector bundles ....................... 22 3.4 In physics language .................................. 24 Notation These notes compile some general facts about connections on principal bundles, and their relation to connections on vector bundles. A few notes on notation: Weare working in the context of a principal 퐺-bundle 푃 over a manifold 휋 푀. This we denote as 퐺 ↪ 푃 → 푀; where 휋 ∶ 푃 → 푀 is the projection map. The right action of 퐺 on 푃 is denoted as 휎 ∶ 푃 ×퐺 → 푃. For any 푔 ∈ 퐺, we denote right multiplication by 푔 as 휎푔 ∶ 푃 → 푃; and for every 푝 ∈ 푃, we denote the orbit map as 휎푝 ∶ 퐺 → 푃. Given a smooth function 푓 ∶ 푀 → 푁 between manifolds, we denote the tangent map at some 푥 ∈ 푀 as 푇푥푓 ∶ 푇푥푀 → 푇푓(푥)푁. This is to explicitly show the functoriality of 푇푥. 1 Connections on principal bundles 1.1 Connections as horizontal distributions Recall that a vector 푣 ∈ 푇푝푃 is called vertical if 푇푝휋(푣) = 0. We denote the subspace of vertical vectors by 푉푝푃 ⊂ 푇푝푃. By definition, 푉푝푃 is nothing more than the kernel of 푇푝휋, so we have a short exact sequence 푇푝휋 0 푉푝푃 푇푝푃 푇휋(푝)푀 0 . *Please send corrections, suggestions, etc. to [email protected]. Latest version on homotopico.com/notes . 1 Since this is a sequence of vector spaces, it splits, and thus we have an isomorphism 푇푝푃 ≅ 푉푝푃 ⊕ 푇휋(푝)푀. However, the splitting (and thus the isomorphism) is not canonical: it depends on a choice of a sub- space 퐻푝 ⊂ 푇푝푃 that is complementary to 푉푝푃, and an isomorphism 푇휋(푝)푀 → 퐻푝. We call any complementary space to 푉푝푃 a horizontal space at 푝, such that: 푇푝푃 = 푉푝푃 ⊕ 퐻푝. Figure 1: A choice of a horizontal space 퐻푝 at 푇푝푃. There are many such choices (in dotted lines). Once we have chosen a single horizontal subspace 퐻푝 ⊂ 푇푝푃 at 푝, we can find horizontal subspaces for all points in the same fiber of 푝. This follows since the action of 퐺 on 푃, which we denote 휎푔(푝) = 푝 ⋅ 푔, is a fiber-preserving diffeomorphism, and thus 푇푝휎푔 is an isomorphism of tangent spaces that preserves the vertical subspace. This suggests that 푇푝휎푔(퐻푝) is a horizontal subspace at 푝 ⋅ 푔. Indeed, noting that 푇푝⋅푔휋 ∘ 푇푝휎푔 = 푇푝(휋 ∘ 휎푔) = 푇푝휋(푣), we see that 푇푝휋(푉푝푃) ⊆ 푉푝⋅푔푃. Similarly, if 푢 ∈ 푉푝⋅푔푃, we can write 푢 = 푇푝휎푔(푇푝⋅푔휎푔−1 (푢)) = 푇푝휎푔( ̃푢), where by the same argument above ̃푢= 푇푝⋅푔휎푔−1 (푢) ∈ 푉푝푃 is vertical. Therefore, we obtain that 푉푝⋅푔푃 = 푇푝휎푔(푉푝푃). Furthermore, since 푇푝휎푔 ∶ 푇푝푃 → 푇푝⋅푔푃 is an isomorphism, we obtain that 푇푝⋅푔푃 = 푇푝휎푔(푇푝푃) = 푇푝휎푔(푉푝푃) ⊕ 푇푝휎푔(퐻푝) = 푉푝⋅푔푃 ⊕ 푇푝휎푔(퐻푝), And so we have proved the following: Lemma 1.1 (Translation of horizontal subspaces). If 퐻푝 ⊂ 푇푝푃 is horizontal at 푝, then for all 푔 ∈ 퐺, 푇푝휎푔(퐻푝) is horizontal at 푝 ⋅ 푔. So far we have been working at a single point 푝 ∈ 푃. We can now consider a smooth choice of horizontal spaces above each element of 푃: Definition 1.2 ((Principal) Connection). A connection or Ehresmann connection on 푃 is a distribution 퐻 on 푃 such that for all 푝 ∈ 푃, 퐻푝 ⊂ 푇푝푃 is a horizontal subspace. We say that a connection 퐻 is principal if it is compatible with the group action in the sense that for all 푔 ∈ 퐺 and all 푝 ∈ 푃, 푇푝휎푔(퐻푝) = 퐻푝⋅푔. 2 The notion of connection is independent of the group action on the total space 푃, and indeed it ap- plies to general fiber bundles. The condition for a connection to be principal states that our choice of horizontal subspaces along a single fiber is consistent with the “translation” lemma 1.1. We think of a connection 퐻 as a preferred way of relating “neighboring” fibers of the bundle. Once we have 푝 ∈ 푃, we might think that the preferred way of moving to another fiber is along a “direction” (i.e. tangent vector) in the horizontal space 퐻푝. This gives us a little bit of intuition and (sort of) justifies (kind of) the name connection. In practice, however, working with distributions might be cumbersome. Fortunately for us, there are other (equivalent) presentations of connections. 1.2 Connections as 1-forms Let 픤 be the Lie algebra of 퐺. Recall that for all 푝 ∈ 푃, we have the infinitesimal action of 픤 on 푇푝푃, 푎푝 ∶ 픤 → 푇푝푃 given as d | 푎푝(푋) ∶= | 푝 ⋅ exp(푡푋). d푡 |푡=0 Writing 휎푝 ∶ 퐺 → 푃 as 휎푝(푔) = 푝 ⋅ 푔, we see that the infinitesimal action is simply the differential of 휎푝: 푎푝(푋) = 푇푒휎푝(푋). This infinitesimal action induces, for each 푋 ∈ 픤, a vector field 푋♯ called the f undamental vector field associated to 푋 given by ♯ 푋푝 ∶= 푎푝(푋). We have that 휎푝 is a diffeomorphism onto the fiber containing 푝, and thus 푎푝 = 푇푒휎푝 induces a 푎푝 linear isomorphism 픤 ≅ 푉푝푃. Suppose that we have a principal connection 퐻 on 푃. Then in particular, we have a subspace 퐻푝 ⊂ 푇푝푃 such that 푇푝푃 = 푉푝푃 ⊕ 퐻푝, and so we can construct a map 휔푝 ∶ 푇푝푃 → 픤 as 푉 퐻 −1 푉 휔푝(푣 + 푣 ) = 푎푝 (푣 ), 푉 퐻 where 푣 ∈ 푉푝푃 and 푣 ∈ 퐻푝. By construction, we have that 휔푝(푎푝(푋)) = 푋 for all 푋 ∈ 픤. Wecan also see how 휔푝 compares to 휔푝⋅푔, since we know that our horizontal distribution behaves nicely along the fibers of the action. For this, first note that for all 푔 ∈ 퐺, d | 푇푝휎푔(푎푝(푋)) = | 휎푔(푝 ⋅ exp(푡푋)) d푡 |푡=0 d | = | 푝 ⋅ exp(푡푋)푔 d푡 |푡=0 d | = | (푝 ⋅ 푔) ⋅ (푔−1 exp(푡푋)푔). d푡 |푡=0 Now we ask ourselves, do we know what the tangent vector of 푔−1 exp(푡푋)푔 is? Yes, yes we do: d | −1 d | | 푔 exp(푡푋)푔 = | Conj푔−1 (exp(푡푋)) = Ad푔−1 (푋), d푡 |푡=0 d푡 |푡=0 1 −1 where we have written Conj푔(ℎ) = 푔ℎ푔 , and Ad푔 = 푇푒 Conj푔. Then we have d | −1 푇푝휎푔(푎푝(푋)) = | (푝 ⋅ 푔) ⋅ (푔 exp(푡푋)푔) = 푎푝⋅푔(Ad푔−1 (푋)). d푡 |푡=0 푉 퐻 푉 With this, we can see that for 푣 ∈ 푇푝푃, which we write as 푣 = 푣 + 푣 with 푣 = 푎푝(푋) for some 푋 ∈ 픤: ∗ 푉 퐻 푉 퐻 (휎푔 휔)푝(푣 +푣 ) = 휔푝⋅푔(푇푝휎푔(푣 )+푇푝휎푔(푣 )) = 휔푝⋅푔(푇푔휎푔(푎푝(푋))) = Ad푔−1 (푋) = (Ad푔−1 ∘휔푝)(푣), 1https://xkcd.com/927/ 3 and so we conclude that ∗ (휎푔 휔) = Ad푔−1 ∘휔. Then we have proved, modulo the small detail of smoothness2, the following: 1 Proposition 1.3 ( -form induced by principal휋 connection). Let 퐻 be a principal connection on 퐺 ↪ 푃 → 푀. Then there exists a (unique) 픤-valued 1-form 휔 ∈ Ω1(푃, 픤), such that for all 푝 ∈ 푃, 푔 ∈ 퐺 and 푋 ∈ 픤: 1. 휔푝(푎푝(푋)) = 푋, ∗ 2. 휎푔 휔 = Ad푔−1 ∘휔, and 3. ker(휔푝) = 퐻푝. We call any 픤-valued 1-form satisfying these properties a connection 1-form: Definition 1.4 (Connection 1-form). A connection 1-form on 푃 is a 픤-valued 1-form 휔 ∈ Ω1(푃, 픤) such that for all 푝 ∈ 푃, 푔 ∈ 퐺 and 푋 ∈ 픤: 1. 휔푝(푎푝(푋)) = 푋, and ∗ 2. 휎푔 휔 = Ad푔−1 ∘휔. The converse to proposition 1.3 is also true: Proposition 1.5. Principal connection induced by connection 1-form Let 휔 ∈ Ω1(푃, 픤) be a connection 1-form. Then the distribution 퐻 defined pointwise as 퐻푝 = ker(휔푝) ⊂ 푇푝푃 is a principal connection on 푃. Proof. — First, let’s see that indeed 퐻푝 = ker(휔푝) is horizontal. If 푣 ∈ ker(휔푝)∩푉푝푃, then 푣 = 푎푝(푋) for some 푋 ∈ 픤, so that 0 = 휔푝(푣) = 휔푝(푎푝(푋)) = 푋, and thus 푣 = 0. Therefore ker(휔푝) ∩ 푉푝푃 = {0}. Now for an arbitrary 푣 ∈ 푇푝푃, set 푉 푣 = 푎푝(휔푝(푣)). 푉 푉 Then we have that 푇푝휋(푣 ) = 0, since it is in the image of 푎푝, and thus 푣 ∈ 푉푝푃. Finally, setting 푣퐻 = 푣 − 푣푉 , we have 퐻 휔푝(푣 ) = 휔푝(푣) − 휔푝(푎푝(휔푝(푣))) = 휔푝(푣) − 휔푝(푣) = 0, 퐻 푉 퐻 푉 퐻 and so 푣 ∈ ker 휔푝 = 퐻푝. We have then shown that 푣 = 푣 + 푣 , with 푣 ∈ 푉푝푃 and 푣 ∈ 퐻푝, and so 푇푝푃 = 푉푝푃 ⊕ 퐻푝. Thus 퐻푝 is a horizontal subspace. Now to see that 퐻 is principal, note that 휔푝⋅푔(푇푝휎푔(푣)) = Ad푔−1 (휔푝(푣)). Since both 푇푝휎푔 and Ad푔−1 are isomorphisms, we have that 푣 ∈ ker 휔푝 if and only if 푇푝휎푔(푣) ∈ ker 휔푝⋅푔, and thus 푇푝휎푔(퐻푝) = 퐻푝⋅푔. Finally, smoothness follows from the fact that 휔 is a smooth form. ■ From now on, if 휔 is a connection 1-form, we will simply call it a connection.
Load more

Recommended publications
  • 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]
  • Cartan Connections to Connections on fibre Bundles, and Some Modern Applications
    The works of Charles Ehresmann on connections: from Cartan connections to connections on fibre bundles, and some modern applications Charles-Michel Marle Universite´ Pierre et Marie Curie Paris, France The works of Charles Ehresmann on connections: from Cartan connections to connections on fibre bundles, and some modern applications – p. 1/40 Élie Cartan’s affine connections (1) Around 1923, Élie Cartan [1, 2, 3] introduced the notion of an affine connection on a manifold. That notion was previously used, in a less general setting, by H. Weyl [16] and rests on the idea of parallel transport due to T. Levi-Civita [11]. The works of Charles Ehresmann on connections: from Cartan connections to connections on fibre bundles, and some modern applications – p. 2/40 Élie Cartan’s affine connections (1) Around 1923, Élie Cartan [1, 2, 3] introduced the notion of an affine connection on a manifold. That notion was previously used, in a less general setting, by H. Weyl [16] and rests on the idea of parallel transport due to T. Levi-Civita [11]. A large part of [1, 2] is devoted to applications of affine connections to Newtonian and Einsteinian Mechanics. Cartan show that the principle of inertia (which is at the foundations of Mechanics), according to which a material point particle, when no forces act on it, moves along a straight line with a constant velocity, can be expressed locally by the use of an affine connection. Under that form, that principle remains valid in (curved) Einsteinian space-times. The works of Charles Ehresmann on connections: from Cartan connections to connections on fibre bundles, and some modern applications – p.
    [Show full text]
  • Pseudogroups Via Pseudoactions: Unifying Local, Global, and Infinitesimal Symmetry
    PSEUDOGROUPS VIA PSEUDOACTIONS: UNIFYING LOCAL, GLOBAL, AND INFINITESIMAL SYMMETRY ANTHONY D. BLAOM Abstract. A multiplicatively closed, horizontal foliation on a Lie groupoid may be viewed as a `pseudoaction' on the base manifold M. A pseudoaction generates a pseudogroup of transformations of M in the same way an ordinary Lie group action generates a transformation group. Infinitesimalizing a pseu- doaction, one obtains the action of a Lie algebra on M, possibly twisted. A global converse to Lie's third theorem proven here states that every twisted Lie algebra action is integrated by a pseudoaction. When the twisted Lie algebra action is complete it integrates to a twisted Lie group action, according to a generalization of Palais' global integrability theorem. Dedicated to Richard W. Sharpe 1. Introduction 1.1. Pseudoactions. Unlike transformation groups, pseudogroups of transfor- mations capture simultaneously the phenomena of global and local symmetry. On the other hand, a transformation group can be replaced by the action of an ab- stract group which may have nice properties | a Lie group in the best scenario. Here we show how to extend the abstraction of group actions to handle certain pseudogroups as well. These pseudogroups include: (i) all Lie pseudogroups of finite type (both transitive and intransitive) and hence the isometry pseudogroups of suitably regular geometric structures of finite type; (ii) all pseudogroups gen- erated by the local flows of a smooth vector field; and, more generally (iii) any pseudogroup generated by the infinitesimal action of a finite-dimensional Lie al- gebra. Our generalization of a group action on M, here called a pseudoaction, consists of a multiplicatively closed, horizontal foliation on a Lie groupoid G over arXiv:1410.6981v4 [math.DG] 3 Oct 2015 M.
    [Show full text]
  • Bulletin De La S
    BULLETIN DE LA S. M. F. F. BRICKELL R.S. CLARK Conformally riemannian structures. I Bulletin de la S. M. F., tome 90 (1962), p. 15-26 <http://www.numdam.org/item?id=BSMF_1962__90__15_0> © Bulletin de la S. M. F., 1962, tous droits réservés. L’accès aux archives de la revue « Bulletin de la S. M. F. » (http: //smf.emath.fr/Publications/Bulletin/Presentation.html) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/ conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright. Article numérisé dans le cadre du programme Numérisation de documents anciens mathématiques http://www.numdam.org/ Dull. Soc. math. France^ 90, 1962, p. i5 a 26. CONFORMALLY RIEMANNIAN STRUCTURES, I ; F. RRICKELL AND R. S. CLARK (Southampton). Introduction. — We define a conformally Riemannian structure on a differentiable (1) manifold M of dimension n to be a differentiable subor- dinate structure of the tangent bundle to M whose group G consists of the non-zero scalar multiples of the orthogonal n x n matrices. The method of equivalence of E. CARTAN [I], as described by S. CHERN [3], associates with a given conformal structure a certain principal fibre bundle on which a set of linear differential forms is denned globally. We obtain such a bundle and set of forms explicitly and show their relation to the normal conformal connection of E. CARTAN | 2]. The first paragraph contains an exposition of conformal connections in tlie light of C.
    [Show full text]
  • Elementary Differential Geometry
    ELEMENTARY DIFFERENTIAL GEOMETRY YONG-GEUN OH { Based on the lecture note of Math 621-2020 in POSTECH { Contents Part 1. Riemannian Geometry 2 1. Parallelism and Ehresman connection 2 2. Affine connections on vector bundles 4 2.1. Local expression of covariant derivatives 6 2.2. Affine connection recovers Ehresmann connection 7 2.3. Curvature 9 2.4. Metrics and Euclidean connections 9 3. Riemannian metrics and Levi-Civita connection 10 3.1. Examples of Riemannian manifolds 12 3.2. Covariant derivative along the curve 13 4. Riemann curvature tensor 15 5. Raising and lowering indices and contractions 17 6. Geodesics and exponential maps 19 7. First variation of arc-length 22 8. Geodesic normal coordinates and geodesic balls 25 9. Hopf-Rinow Theorem 31 10. Classification of constant curvature surfaces 33 11. Second variation of energy 34 Part 2. Symplectic Geometry 39 12. Geometry of cotangent bundles 39 13. Poisson manifolds and Schouten-Nijenhuis bracket 42 13.1. Poisson tensor and Jacobi identity 43 13.2. Lie-Poisson space 44 14. Symplectic forms and the Jacobi identity 45 15. Proof of Darboux' Theorem 47 15.1. Symplectic linear algebra 47 15.2. Moser's deformation method 48 16. Hamiltonian vector fields and diffeomorhpisms 50 17. Autonomous Hamiltonians and conservation law 53 18. Completely integrable systems and action-angle variables 55 18.1. Construction of angle coordinates 56 18.2. Construction of action coordinates 57 18.3. Underlying geometry of the Hamilton-Jacobi method 61 19. Lie groups and Lie algebras 62 1 2 YONG-GEUN OH 20. Group actions and adjoint representations 67 21.
    [Show full text]
  • Math 704: Part 1: Principal Bundles and Connections
    MATH 704: PART 1: PRINCIPAL BUNDLES AND CONNECTIONS WEIMIN CHEN Contents 1. Lie Groups 1 2. Principal Bundles 3 3. Connections and curvature 6 4. Covariant derivatives 12 References 13 1. Lie Groups A Lie group G is a smooth manifold such that the multiplication map G × G ! G, (g; h) 7! gh, and the inverse map G ! G, g 7! g−1, are smooth maps. A Lie subgroup H of G is a subgroup of G which is at the same time an embedded submanifold. A Lie group homomorphism is a group homomorphism which is a smooth map between the Lie groups. The Lie algebra, denoted by Lie(G), of a Lie group G consists of the set of left-invariant vector fields on G, i.e., Lie(G) = fX 2 X (G)j(Lg)∗X = Xg, where Lg : G ! G is the left translation Lg(h) = gh. As a vector space, Lie(G) is naturally identified with the tangent space TeG via X 7! X(e). A Lie group homomorphism naturally induces a Lie algebra homomorphism between the associated Lie algebras. Finally, the universal cover of a connected Lie group is naturally a Lie group, which is in one to one correspondence with the corresponding Lie algebras. Example 1.1. Here are some important Lie groups in geometry and topology. • GL(n; R), GL(n; C), where GL(n; C) can be naturally identified as a Lie sub- group of GL(2n; R). • SL(n; R), O(n), SO(n) = O(n) \ SL(n; R), Lie subgroups of GL(n; R).
    [Show full text]
  • Notes on Principal Bundles and Classifying Spaces
    Notes on principal bundles and classifying spaces Stephen A. Mitchell August 2001 1 Introduction Consider a real n-plane bundle ξ with Euclidean metric. Associated to ξ are a number of auxiliary bundles: disc bundle, sphere bundle, projective bundle, k-frame bundle, etc. Here “bundle” simply means a local product with the indicated fibre. In each case one can show, by easy but repetitive arguments, that the projection map in question is indeed a local product; furthermore, the transition functions are always linear in the sense that they are induced in an obvious way from the linear transition functions of ξ. It turns out that all of this data can be subsumed in a single object: the “principal O(n)-bundle” Pξ, which is just the bundle of orthonormal n-frames. The fact that the transition functions of the various associated bundles are linear can then be formalized in the notion “fibre bundle with structure group O(n)”. If we do not want to consider a Euclidean metric, there is an analogous notion of principal GLnR-bundle; this is the bundle of linearly independent n-frames. More generally, if G is any topological group, a principal G-bundle is a locally trivial free G-space with orbit space B (see below for the precise definition). For example, if G is discrete then a principal G-bundle with connected total space is the same thing as a regular covering map with G as group of deck transformations. Under mild hypotheses there exists a classifying space BG, such that isomorphism classes of principal G-bundles over X are in natural bijective correspondence with [X, BG].
    [Show full text]
  • Cartan Connection Applied to Dynamic Calculation in Robotics
    Cartan connection applied to dynamic calculation in robotics Diego Colón, 1 Phone +55-11-3091-5650 Email [email protected] 1 Laboratório de Automação e Controle - LAC/PTC, Escola Politécnica da Universidade de São Paulo, São Paulo, Brazil Received: 31 January 2017 / Accepted: 6 June 2018 Abstract A Cartan connection is an important mathematical object in differential geometry that generalizes, to an arbitrary Riemannian space, the concept of angular velocity (and twists) of a non-inertial reference frame. In fact, this is an easy way to calculate the angular velocities (and twists) in systems with multiple reference frames, like a robot or a complex mechanism. The concept can also be used in different representations of rotations and twists, like in the Lie groups of unit quaternions and unit dual quaternions. In this work, the Cartan connection is applied to the kinematic and dynamic calculations of a systems of with multiple reference frames that could represent a robotic serial chain or mechanism. Differently from previous works, the transformation between frames is represented as unit quaternions, which are known to be better mathematical representations from the numerical point of view. It is also generalized to this new quaternion representation, previous results like the extended Newton’s equation, the covariant derivative and the Cartan connection. An example of application is provided. AQ1 Keywords Quaternions Robotics Lie groups Screw theory Technical Editor: Victor Juliano De Negri. 1. Introduction Concepts of differential geometry and Lie groups and Lie algebras have been used in many areas of engineering for a long time. More recently, a renewed interest could be verified in mechanism and robotics [1, 2, 10, 16, 23], where dual quaternions were used, and very complete reference books, like [3], appeared.
    [Show full text]
  • Complete Connections on Fiber Bundles
    Complete connections on fiber bundles Matias del Hoyo IMPA, Rio de Janeiro, Brazil. Abstract Every smooth fiber bundle admits a complete (Ehresmann) connection. This result appears in several references, with a proof on which we have found a gap, that does not seem possible to remedy. In this note we provide a definite proof for this fact, explain the problem with the previous one, and illustrate with examples. We also establish a version of the theorem involving Riemannian submersions. 1 Introduction: A rather tricky exercise An (Ehresmann) connection on a submersion p : E → B is a smooth distribution H ⊂ T E that is complementary to the kernel of the differential, namely T E = H ⊕ ker dp. The distributions H and ker dp are called horizontal and vertical, respectively, and a curve on E is called horizontal (resp. vertical) if its speed only takes values in H (resp. ker dp). Every submersion admits a connection: we can take for instance a Riemannian metric ηE on E and set H as the distribution orthogonal to the fibers. Given p : E → B a submersion and H ⊂ T E a connection, a smooth curve γ : I → B, t0 ∈ I, locally defines a horizontal lift γ˜e : J → E, t0 ∈ J ⊂ I,γ ˜e(t0)= e, for e an arbitrary point in the fiber. This lift is unique if we require J to be maximal, and depends smoothly on e. The connection H is said to be complete if for every γ its horizontal lifts can be defined in the whole domain. In that case, a curve γ induces diffeomorphisms between the fibers by parallel transport.
    [Show full text]
  • Smoothing Maps Into Algebraic Sets and Spaces of Flat Connections
    SMOOTHING MAPS INTO ALGEBRAIC SETS AND SPACES OF FLAT CONNECTIONS THOMAS BAIRD AND DANIEL A. RAMRAS n Abstract. Let X ⊂ R be a real algebraic set and M a smooth, closed manifold. We show that all continuous maps M ! X are homotopic (in X) to C1 maps. We apply this result to study characteristic classes of vector bundles associated to continuous families of complex group representations, and we establish lower bounds on the ranks of the homotopy groups of spaces of flat connections over aspherical manifolds. 1. Introduction The first goal of this paper is to prove the following result about the differential topology of algebraic sets. Theorem 1.1 (Section2) . Let X ⊂ Rn be a (possibly singular) real algebraic set, and let f : M ! X be a continuous map from a smooth, closed manifold M. Then there exists a map g : M ! X, and a homotopy H : M × I ! X connecting f and g g, such that the composite M ! X,! Rn is C1. The problem of smoothing maps into algebraic sets seems natural, but we have not found mention of it in the literature. We consulted several experts in real algebraic geometry; some expected our result to hold, and some did not. Our proof proceeds by embedding X as the singular set of an irreducible, quasi- projective variety Y and using a resolution of singularities Ye ! Y for which the inverse image of X is a divisor with normal crossing singularities. Basic facts about neighborhoods of algebraic sets then reduce the problem to the case of normal crossing divisors, which can be handled by differential-geometric means.
    [Show full text]
  • WHAT IS a CONNECTION, and WHAT IS IT GOOD FOR? Contents 1. Introduction 2 2. the Search for a Good Directional Derivative 3 3. F
    WHAT IS A CONNECTION, AND WHAT IS IT GOOD FOR? TIMOTHY E. GOLDBERG Abstract. In the study of differentiable manifolds, there are several different objects that go by the name of \connection". I will describe some of these objects, and show how they are related to each other. The motivation for many notions of a connection is the search for a sufficiently nice directional derivative, and this will be my starting point as well. The story will by necessity include many supporting characters from differential geometry, all of whom will receive a brief but hopefully sufficient introduction. I apologize for my ungrammatical title. Contents 1. Introduction 2 2. The search for a good directional derivative 3 3. Fiber bundles and Ehresmann connections 7 4. A quick word about curvature 10 5. Principal bundles and principal bundle connections 11 6. Associated bundles 14 7. Vector bundles and Koszul connections 15 8. The tangent bundle 18 References 19 Date: 26 March 2008. 1 1. Introduction In the study of differentiable manifolds, there are several different objects that go by the name of \connection", and this has been confusing me for some time now. One solution to this dilemma was to promise myself that I would some day present a talk about connections in the Olivetti Club at Cornell University. That day has come, and this document contains my notes for this talk. In the interests of brevity, I do not include too many technical details, and instead refer the reader to some lovely references. My main references were [2], [4], and [5].
    [Show 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.
    [Show full text]