Note that for a, b G, we have L a R b = R b L a, by the associative law. We'll use e to denote the identity element of G.

Transcription

1 NOTES ON PRINCIPAL BUNDLES, CONNECTIONS, AND HOMOGENEOUS SPACES LANCE D. DRAGER 1. Introduction These notes are meant to accompany talks in the Geometry Seminar at Texas Tech during the Spring Semester of 29. These notes will be updated frequently, so it's best to keep track of them on the web. Note the version time stamp at the bottom of the rst page. At this point, the seminar participants should be comfortable with the basic dierential geometry apparatus done from the point of view of principal bundles. I hope to come back and add this material to these notes at some point in the future. Thanks to the seminar participants for listening to me and straightening me out. Any errors are, of course, my own. 2. Group Actions Let G be a Lie Group. If g G we have the multiplication maps L g : G G: x gx R g : G G: x xg. Note that for a, b G, we have L a R b = R b L a, by the associative law. We'll use e to denote the identity element of G. Version Time-stamp: " :24:36 drager". clance D. Drager. BY: $\ = This work is licensed under the Creative Commons Attribution- Noncommercial-No Derivative Works 3. United States License. To view a copy of this license, visit or send a letter to Creative Commons, 171 Second Street, Suite 3, San Francisco, California, 9415, USA.. 1

2 2 LANCE D. DRAGER We identify g, the Lie Algebra of G, with T e G, the tangent space at the identity. If A g, we denote the left invariant vector eld generated by A by λ(a), and we denote the value of this vector eld at g G by λ g (A). Thus, we have (2.1) λ g (A) = (L g ) A T g G. where is the tangent map of L g. (L g ) : T e G T g G Notation 2.1. In what follows we will usually drop the lower star when writing the tangent map of L g or R g. Thus, we will usually write the equation (2.1) as just λ g (A) = L g A. This simplies the notation and there is usually no danger of confusion, since there is not much else that L g A could mean. The Lie bracket of two left invariant vector elds is left invariant, and we dene the bracket on g by Thus, in general, [A, B] := [λ(a), λ(a)] e. [λ(a), λ(b)] = λ([a, B]) We denote the right invariant vector eld generated by A g by ρ(a). Since we used left invariant vector elds to dene the bracket in g, we have for right invariant vector elds and so [ρ(a), ρ(b)] e = [A, B] [ρ(a), ρ(b)] = ρ([a, B]).

3 PRINCIPAL BUNDLES, CONNECTIONS AND HOMOGENEOUS SPACES 3 For A g we write exp(ta) = e ta for the exponential map of G. We have d dt R e ta(g) = d t= dt ge ta t= = d dt L g (e ta ) = L g A = λ g (A). Thus the ow {ϕ t } of λ(a) is ϕ t = R e ta. Similarly, the ow {ψ t } of ρ(a) is ψ t = L e ta. The following observations will be useful in a moment. Let M and N be manifolds. Let π 1 : M N M and π 2 : M N N be the projections. Recall that one usually identies T (p,q) (M N) with T p M T q N via the map T (p,q) (M N) X ((π 1 ) X, (π 2 ) X) T p M T q N. Given a vector eld X on M, we can dene a vector eld ˆX on M N by ˆX (p,q) = (X p, q ). We can characterize ˆX as the unique vector eld that is π 1 -related to X and π 2 -related to the zero vector eld. If X 1 and X 2 are vector elds on M, we see that [ ˆX 1, ˆX 2 ] is π 1 - related to [X 1, X 2 ] and π 2 -related to [, ] =. Thus, [ ˆX 1, ˆX 2 ] = [X 1, X 2 ] Now suppose that G acts on M on the left. Let A be an element of the Lie Algebra g. We can dene a ow {ϕ t } on M by ϕ t (p) = e ta p. This ow comes from a vector eld, which we will denote by ρ M (A). Thus, ρ M p (A) = d dt e ta p. We now have a map A ρ M (A) from g to the Lie algebra of vector elds on M. Naturally, we want to know what [ρ M (A), ρ M (B)] is. The answer is [ρ M (A), ρ M (B)] = ρ M ([A, B]). To see this, let m: G M M : (g, p) gp

4 4 LANCE D. DRAGER be the group action. Given A g, we have the right invariant vector eld ρ(a) on G, and so we get the vector eld ˆρ(A) on G M as above. We have m (ˆρ (g,p) (A)) = m (ρ g (A), p ) = d dt m(e ta g, p) = d dt e ta gp = ρ M gp(a). Thus, ρ M (A) is the unique vector eld on M that is m-related to ˆρ(A). If B is also in g, then [ˆρ(A), ˆρ(B)] is m-related to [ρ M (A), ρ M (B)]. But [ˆρ(A), ˆρ(B)] = [ρ(a), ρ(b)] = ˆρ([A, B]). Next, we consider the action on these vector elds of equivariant maps. Proposition 2.2. Let G act on the left of manifolds M and N and let f : M N be an equivariant map, i.e., f(gp) = gf(p). The we have f (ρ M p (A)) = ρ N f(p)(a), A g. In other words, ρ M (A) is f-related to ρ N (A). Proof. This is a simple calculation: f (ρ M p (A)) = d dt f(e ta p) = d dt e ta f(p) = ρ N f(p)(a). We can also see the eect of a group translation on ρ M (A). Proposition 2.3. Let G act on M on the left. Then, for g G and A g, we have L g ρ M x (A) = ρ M gx(ad(g)a), where Ad is the adjoint representation of G.

5 PRINCIPAL BUNDLES, CONNECTIONS AND HOMOGENEOUS SPACES 5 Proof. Just calculate: L g ρ M x (A) = d dt L g (e ta x) = d dt ge ta x = d dt ge ta g 1 gx = d dt e t Ad(g)A gx = ρ M gx(ad(g)a). For a xed p M, we have the map ρ M p ( ): g T p M. We want to know something about the kernel and the image of this map. For p in M, let H p be the isotropy group of p, i.e., H p := {g G gp = p}. This is a closed subgroup of G. The Lie algebra of H p will be denoted by h p. Proposition 2.4. The kernel of ρ M p ( ): g T p M is h p. Proof. If B h p, then e tb H p of all t. Thus, (2.2) e tb p = p. Dierentiating this equation at t = gives ρ M p (B) =. Conversely, suppose that ρ M p (B) =. This means that p is a zero of the vector eld ρ M (B), as so p is an equilibrium point of the ow. This means that (2.2) holds for all t. Thus, e tb H p for all t, and this implies that B h p. The main result we need about the image of this map is the following. Proposition 2.5. If G act transitively on M, the map ρ M p ( ): g T p M is surjective. This follows from an important theorem discussed earlier in the seminar.

6 6 LANCE D. DRAGER Theorem 2.6. Let G act transitively on M. Fix p M and dene π G : G M : g gp. Then π G : G M has the structure of a principal H p bundle. In particular, π G is a submersion. To get Proposition 2.5, note that the last theorem implies that (π G ) : g T p M is surjective. But (π G ) (A) = d dt π G (e ta ) = d dt e ta p = ρ M p (A). Remark 2.7. In general, ρ M p (g) is the tangent space to the orbit of p, which is an immersed submanifold. We can, of course, repeat these constructions with right actions. We merely record the results. Proposition 2.8. Suppose that G acts on the right of M. (1) For each A g, we get a vector eld λ M (A) on M by λ M p (A) = d dt pe ta (2) We have (3) We have [λ M (A), λ M (A)] = λ M ([A, B]) R g λ M p (A) = λ M pg(ad(g 1 )A). The results about the kernel and image are the same. 3. General Invariant Connections In this section we study invariant connections on a homogeneous space, without assuming any special structure on the homogeneous space. Later we will consider the reductive case.

7 PRINCIPAL BUNDLES, CONNECTIONS AND HOMOGENEOUS SPACES The Basic Setup. We begin with some notation. Notation 3.1. If V is a vector space, GL(V ) denotes the Lie Group of linear automorphisms of G. The Lie algebra of GL(V ) will be identi- ed, in the usual way, with gl(v ), the space of linear endomorphisms of V, with the commutator bracket. We want to study homogeneous spaces, so we assume the following setup. Let M be a manifold and assume that a Lie group G acts transitively on the left of M. Fix a base point p in M, and let H = H p be the isotropy group at p. We choose a model space V for the tangent spaces of M and we x a particular isomorphism u : V T p M. Later on we will have natural choices for V and u, but we won't specify them yet. If h H, we have L h p = p, so we get an induced map T po L h T p M T po M. The mapping H GL(T p M): h T p L h is a representation of H called the Isotropy Representation. Since u is an isomorphism, we get an induced map α(h): V V so that the following diagram commutes (3.1). α(h) V V u u T p M T p M T p L h Thus, we get a representation α: H GL(V ), which is equivalent to the isotropy representation. Just to have a name for it, let's call α the Model Representation. We now want to form the bundle of V -frames of M. A V -frame at p M is, by denition, a linear isomorphism u: V T p M.

8 8 LANCE D. DRAGER The set of V -frames at p is denoted by F p (M). The group GL(V ) acts transitively and freely on the right of F p (M) by composition, i.e. ua := u a, as in the following commutative diagram. a V V u T p M ua We set F (M) = p F p(m) and dene a projection π : F (M) M by π(f p (M)) = {p}. As discussed in the seminar, F (M) π M has the structure of a GL(V )-principal bundle. Recall that the vertical space V u T u F (M) is the tangent space to the ber. It can also be described as the kernel of T u π. The group GL(V ) acts on the right of F (M) and the orbits are exactly the bers. As in our discussion of group actions, an element A gl(v ) gives us a vertical vector eld V u (A) = d dt ue ta. We used λ F u (M) (A) for this in the last section, but we'll use V u (A) in this case to remind us that it's vertical. Since this vector eld comes from a right action, we have (3.2) R a V u (A) = V ua (Ad(a 1 )A). Since the action of GL(V ) is free (i.e., no isotropy) the map A V u (A) is an isomorphism gl(v ) V u (to see that it's surjective, use a local trivialization). Since G acts on the left of M, we can dene a left action of G on F (M). Suppose g G; since L g is a dieomorphism of M, it induces an isomorphism from T p M to T gp M. If u F p (M), we dene

9 PRINCIPAL BUNDLES, CONNECTIONS AND HOMOGENEOUS SPACES 9 gu = L g u, as in the following commutative diagram. V u T p M gu T gp M T pl g Note that π(gu) = gπ(u), i.e., π is an equivariant map. We now have two group actions on F (M), one on the left and one on the right. Note that for u F (M), g G and a GL(V ) we have (gu)a = g(ua), just because composition of maps is associative. written as This fact can be (3.3) R a L g = L g R a, g G, a GL(V ), so we can say that the actions commute. Since G acts on the left of M, every x g gives us a vector eld ρ M (x) on M. Since G acts on the left of F (M), we have a similar vector eld ρ F (M) (x). Since π is an equivariant map, ( ) (π) ρ F (M) (x) = ρ M π(u)(x). u Notation 3.2. Usually we'll drop the superscripts on the vector elds ρ M (x), etc. It should be clear from context where the vector eld lives. Since the vector elds ρ(x) come from a left action, we have L g ρ p (x) = ρ gp (Ad(g)x) for p M, and a similar equation on F (M). We can interpret (3.3) as saying that R a is an equivariant map with respect to the action of G, so R a ρ u (x) = ρ ua (x). Similarly, we can interpret (3.3) as saying that L g is equivariant with respect to the action of GL(V ), so (3.4) L g V u (A) = V gu (A).

10 1 LANCE D. DRAGER Note that in the diagram(3.1), the composition T p L h u is what we've dened to be hu, and that the composition u α(h) is the right action of α(h) GL(V ) on u. Thus, we have the formula (3.5) hu = u α(h), h H. We've boxed this formula because we will use it frequently. Here is the innitesimal version. Proposition 3.3. Suppose that x g. Then ρ u (x) is vertical if and only if x h. In this case, ρ u (x) = V u (α (x)), where α : h gl(v ) is the Lie algebra homomorphism induced by the model representation α: H GL(V ). Proof. We know that ρ u (x) is vertical if and only if π ρ u (x) =. But π ρ u (x) = ρ p (x) and the kernel of ρ p ( ) is h. If x h, we have ρ u (x) = d dt e tx u = d dt u α(e tx ) = d dt u e tα (x) = V u (α (x)). This completes the description of the basic setting we will use. We next consider invariant connections Invariant Connections. Definition 3.4. Recall that a connection on F (M) can be specied as a distribution H on F (M) such that (1) T F (M) = H V (2) R a H u = H ua for all u F (M) and a GL(V ).

11 PRINCIPAL BUNDLES, CONNECTIONS AND HOMOGENEOUS SPACES 11 There are several ways to dene the notion of a connection being invariant under G, i.e., G acts on M by ane maps. It can be described in terms of parallel translation and covariant derivatives, but the following denition should be plausible. Definition 3.5. We say that H is G-invariant if the action of G preserves H, i.e. L g H u = H gu, g G, u F (M). Now, recall (3.5). If h H, then hu = u α(h), so R α(h 1 )L h u = u. Thus, we get an induced map (3.6) ψ(h) := R α(h 1 )L h : T u F (M) T u F (M) Lemma 3.6. The map ψ : H GL(T u F (M)) is a representation (i.e., a group homomorphism). Proof of Lemma. It's clear that ψ(e) is the identity. Suppose that h and k are in H. Then ψ(hk) = R α((hk) 1 )L hk = R α(k 1 h 1 )L hk = R α(k 1 )α(h 1 )L hk = R α(h 1 )R α(k 1 )L h L k = R α(h 1 )L h R α(k 1 )L k by (3.3), = ψ(h)ψ(k). Lemma 3.7. The vectical space V u is invariant under ψ. Proof of Lemma. We have ψ(h)v u (A) = R α(h 1 )L h V u (A) = R α(h 1 )V hu (A) by (3.4), = R α(h 1 )V u α(h)(a) = V u (Ad(α(h))A) by (3.2).

12 12 LANCE D. DRAGER Suppose now that H is an invariant connection. We then have ψ(h)h u = R α(h 1 )L h H u = R α(h 1 )H hu (H is invariant), = R α(h 1 )H u α(h) = H u. We can go the other way, and construct an invariant connection from a ψ-invariant horizontal space at u. Theorem 3.8. There is a one-to-one correspondence between G- invariant connections on F (M) and subspaces H T u F (M) that satisfy the two conditions (1) T u F (M) = H V u. (2) H is ψ-invariant. In other words, the representation ψ has V u as an invariant subspace, and H is an invariant complement to V u. Proof of Theorem. First suppose that H is an invariant connection. Let u F (M) be an arbitrary point and let p = π(u). Since G acts transitively on M, we can nd g G so that gp = p. Then gu and u are both in F p (M), so we can nd a GL(V ) so that gu a = u. See Figure 1. But then u = R a L g u and R a L g H u = R a H gu = H gu a = H u. Thus, the whole connection H is determined by H u, it's value at u. We know from the discussion above that H u satises properties (1) and and (2) of the theorem. For the converse, suppose that we are give a subspace H with that has properties (1) and (2). We need to dene the horizontal subspace H u at every u F (M). Given u, let p = π(u). We can write u = gu a as before. We would then like to dene (3.7) H u := R a L g H u,

13 PRINCIPAL BUNDLES, CONNECTIONS AND HOMOGENEOUS SPACES 13 u F (M) R a u gu L g π M p p L g Figure 1. u = gu a but we must check that this is independent of our choices. Suppose that we also have u = g u b. Then π(g u b) = g p = p. Thus gp = g p, so we must have g = gh for some h H. We then have gu a = u = g u b = ghu b = gu α(h)b.

14 14 LANCE D. DRAGER Since the action of GL(V ) is free, this implies that α(h)b = a, and so b = α(h 1 )a. But then we have R b L g H = R α(h 1 )al gh H = R a R α(h 1 )L g L h H = R a L g R α(h 1 )L h H = R a L g {ψ(h)h } = R a L g H, since H is ψ-invariant. Hence, we may dene H consistently by (3.7). Since R a L g : T u F (M) T u F (M) is a linear isomorphism that preserves the vertical space, we have T u F (M) = H u V u, from condition (1) of the Theorem. It's also easy to see that H u = H. It remains to check that H has the right invariance properties. To check that H is a connection, we must show R b H u = H ub, b GL(V ). To see this, choose g and a so that gu a = u. Of course ub = gu ab. Then R b H u = R b {R a L g H } = R ab L g H = H gu ab = H ub. To show that H is G-invariant, we must show Writing u = gu a, we have L g H u = H g u, g G. L g H u = L g {R a L g H } = R a L g L g H = R a L g gh = H g gu a = H g u.

15 PRINCIPAL BUNDLES, CONNECTIONS AND HOMOGENEOUS SPACES 15 The proof is now complete. Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX URL: address: