Notes on linear frame bundles Andrew D. Lewis∗ 1995/11/14 The first part of these notes is a review of material from Kobayashi and Nomizu, vol. I, chapter II. We only present that material which is of direct relevance to the material in Chapters III and IV of the same volume. The second part is an overview of selected material from Chapters III and IV. 1. The bundle of linear frames Let M be a manifold of dimension n and let x 2 M.A linear frame at x is an ordered basis, fX1;:::;Xng, of TxM. The collection of all linear frames at x 2 M shall be denoted Lx(M). The collection of all linear frames on M will be denoted L(M) and called the bundle of linear frames or the linear frame bundle. Denote the Lie group of invertible n × n matrices by GL(n; R). This group acts on L(M) on the right by Ra : L(M) ! L(M) i i (x; fX1;:::;Xng) 7! (x; fa1Xi; : : : ; anXig) for a 2 GL(n; R). Heuristically, if we think of fX1;:::;Xng as a column vector, the action looks like 0 1 0 1 X1 X1 B . C t B . C RA @ . A = A @ . A Xn Xn which explains why the action is a right action and not the left actions we see in most cases when we talk about mechanical systems with symmetry. If you are familiar with the concept, you may wish to think of the right action on the linear frame bundle as a \relabelling symmetry." It is readily verified that this action is free (i.e., each orbit is diffeomorphic to the group) and proper (never mind) and so the quotient L(M)=GL(n; R) inherits a manifold structure. The map from L(M)=GL(n; R) to M which assigns to the orbit through u 2 Lx(M) the point x 2 M allows us to identify the orbit space with M. We will denote by π : L(M) ! M the natural projection. So, the bottom line is that π : L(M) ! M is a principal fibre bundle with structure group GL(n; R). To verify that the bundle is locally trivial, we construct a local bundle chart for L(M). Let (U; φ) be a chart for M with coordinates denoted 1 n (x ; : : : ; x ). If u = fX1;:::;Xng is a linear frame at x 2 U, we may write @ @ u = Xi ;:::;Xi : 1 @xi n @xi ∗Professor, Department of Mathematics and Statistics, Queen's University, Kingston, ON K7L 3N6, Canada Email: [email protected], URL: http://www.mast.queensu.ca/~andrew/ 1 2 A. D. Lewis i Xj are the components of an invertible matrix (they map one basis to another) and so the i j −1 coordinates (x ;Xk) establish a diffeomorphism from π (U) to U ×GL(n; R). This verifies i j that L(M) is locally trivial. In fact, the coordinates (x ;Xk) are bundle coordinates for L(M) meaning they enjoy some nice properties with respect to the group action. There is an important, alternative way to think about a linear frame. If we let u = n fX1;:::;Xng be a linear frame at x 2 M, we may regard u as an isomorphism from R to n TxM. If we denote by fe1;:::; eng the standard basis for R , this map is given by n u: R ! TxM i i v ei 7! v Xi: Thought of in this way, what is the right action of GL(n; R) on L(M)? Well, given u 2 n Lx(M) and a 2 GL(n; R), we have to come up with Rau: R ! TxM (i.e., a linear frame at x, in our current thinking). We may regard a as isomorphism on Rn in the natural n n way. It is an exercise to verify that Rau is simply the composition of a: R ! R with n n u: R ! TxM. We will denote by uξ 2 TxM the image of ξ 2 R under u 2 Lx(M). It is easy to think of an element of L(M) as an ordered basis. However, you will also have to be comfortable with regarding u 2 Lx(M) as an isomorphism n from R to TxM since Kobayashi and Nomizu (and hence we) do so without warning. −1 n Let v 2 TxM and u = fX1;:::;Xng 2 Lx(M). One way to think of u v 2 R is to regard it as the components of v in the basis fX1;:::;Xng. 1.1 Example: The easiest example is M = R. Note that GL(1; R) ' R∗, where R∗ is the group of non-zero real numbers with the group operation of multiplication. Any linear frame at x 2 R is given by @ X = a @x for some non-zero number a. To this frame we associate the point (x; a) 2 R × R∗. Thus L(R) ' R×R∗. We may represent L(M) as in Figure1. As coordinates for L(M) ' R×R∗ we shall use (x; a). The action of GL(1; R) in these coordinates appears like ((x; a); b) 7! (x; ab): Now let's examine how we may think of an element of Lx(M) as a linear map from R to TxM. Let ξ 2 R and let u = (x; a) 2 Lx(M). Note that (x; a) represents the frame @ X = a @x . Therefore @ uξ = ξa : • @x 2. Connections in L(M) For A 2 gl(n; R), the Lie algebra of GL(n; R) (i.e., the Lie algebra of n × n matrices), denote by A∗ the fundamental vector field associated with A. In the language our group uses, A∗ is the infinitesimal generator and so is defined by ∗ d A (u) = Rexp(tA)u: dt t=0 Notes on linear frame bundles 3 Lx(M) x M Figure 1. A representation of L(M) when M = R. For u 2 L(M) we define the vertical subspace, Vu(L(M)), at u to be the subspace of Tu(L(M)) generated by all fundamental vector fields at u. This is the same as defining Vu(L(M)) = ker(Tuπ): A connection in L(M) is an assignment of a subbundle, H(L(M)), so that 1. T (L(M)) = H(L(M)) ⊕ V (L(M)), and 2. TuRa(Hu(L(M))) = HRau(L(M)) for each u 2 L(M) and a 2 GL(n; R). A connection in L(M) is called a linear connection. Kobayashi and Nomizu also define affine connections which are connections on the bundle of affine frames. Please ignore these sections in Chapter III. We will reserve the name affine connection for a purpose to be disclosed later (i.e., the one we classically know). Associated with a linear connection is the connection one-form, !, which is a one-form on L(M) which takes its values in gl(n; R). Intrinsically we define the connection one-form by !(u)(X) = fA 2 gl(n; R) j A∗(u) = Xverg: (2.1) Here Xver is the vertical part of X relative to the decomposition given by the connection. The condition2 in the definition of a connection gives the condition !(TuRa(X)) = Ada−1 !(X) (2.2) for u 2 L(M), X 2 Tu(L(M)), and a 2 GL(n; R). Here Ada : gl(n; R) ! gl(n; R) is the image of a 2 GL(n; R) in Autgl(n; R) under the adjoint representation. In fact, any gl(n; R)-valued one-form on P satisfying the conditions (2.1) and (2.2) defines a connection on L(M) given by H(L(M)) = ker(!). We can also define the curvature form on L(M) by Ω(X; Y ) = D!(X; Y ) = d!(Xhor;Y hor) where Xhor and Y hor are the horizontal parts of X and Y , respectively. 4 A. D. Lewis 2.1 Example: We continue the example above of M = R and derive the general form of a linear connection on M. We use coordinates (x; a) for L(M) ' R × R∗. Let's compute the infinitesimal generator for A 2 gl(1; R). The Lie algebra of GL(1; n), gl(1; R), is isomorphic to the Lie algebra R with the zero bracket. For A 2 gl(1; R) ' R, exp(A) = eA 2 R∗ ' GL(1; R). Therefore, ∗ d tA A (x; a) = (x; ae ) = (x; a; 0; aA): dt t=0 @ Therefore, V (L(M)) = h @a i. A complement to this vertical subbundle will be generated by a vector field which is not in the vertical subbundle. Thus, suppose that H(L(M)) is generated by the vector field @ @ h i h(x; a) = R(x; a) + S(x; a) = (x; a; R(x; a);S(x; a)) @x @a where R is non-zero (since the vector field must be complementary to the vertical subbun- dle). The condition of invariance of the horizontal distribution places restrictions on R and S. Let's figure out what those are. If we have h(x; ab) = T(x;a)Rb(h(x; a)) then h will generate a connection. We compute T(x;a)Rb(h(x; a)) = (x; ab; R(x; a); bS(x; a)): This gives R(x; ab) = R(x; a) S(x; ab) = bS(x; a): We may simplify this by supposing that a = 1 and that R = 1. We then see that the horizontal subbundle of a linear connection on R is generated by a vector field of the form @ @ h(x; a) = + aS(x) : @x @a Thus the \slope" of the horizontal subspaces increases as we increase in the positive group direction.