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Chapter 4 Natural Constructions on Vector Bundles

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Chapter 4 Natural Constructions on Vector Bundles 96 CHAPTER 4. NATURAL CONSTRUCTIONS 4.1 Direct sums, tensor products and bun- dles of linear maps Suppose πE : E ! M and πF : F ! M are two vector bundles of rank m and ` respectively, and assume they are either both real (F = R) or both Chapter 4 complex (F = C). In this section we address the following question: given connections on E and F , what natural connections are induced on the other bundles that can be constructed out of E and F ? We answer this by defining parallel transport in the most natural way for each construction Natural Constructions on and observing the consequences. Vector Bundles Direct sums To start with, the direct sum E ⊕ F ! M inherits a natural connection such that the parallel transport of (v; w) 2 E ⊕ F along a path γ(t) 2 M takes the form Contents t t t Pγ (v; w) = (Pγ(v); Pγ(w)): 4.1 Sums, products, linear maps . 96 We leave it as an easy exercise to the reader to verify that the covariant 4.2 Tangent bundles . 100 derivative is then rX (v; w) = (rX v; rX w) 4.2.1 Torsion and symmetric connections . 100 for v 2 Γ(E), w 2 Γ(F ) and X 2 T M. 4.2.2 Geodesics . 103 If E and F have extra structure, this structure is inherited by E ⊕ F . For example, bundle metrics on E and F define a natural bundle metric 4.3 Riemannian manifolds . 105 on E ⊕ F : this has the property that if (e1; : : : ; em) and (f1; : : : ; f`) are orthogonal frames on E and F respectively, then (e1; : : : ; em; f1; : : : ; f`) 4.3.1 The Levi-Civita connection . 105 is an orthogonal frame on E ⊕ F . In fact, E and F need not have the same type of structure: suppose they have structure groups G ⊂ GL(m; F) 4.3.2 Geodesics and arc length . 107 and H ⊂ GL(`; F) respectively. The product G × H is a subgroup of F F F 4.3.3 Computations . 111 GL(m; ) × GL(`; ), which admits a natural inclusion into GL(m + `; ): 3 A 4.3.4 Riemannian submanifolds and surfaces in R . 113 GL(m; F) × GL(`; F) ,! GL(m + `; F) : (A; B) 7! : (4.1) B The combination of G-compatible and H-compatible trivializations on E For the entirety of this chapter, we assume π : E ! M is a smooth and F respectively then gives E⊕F a (G×H)-structure. If the connections vector bundle. We have already seen that any extra structure attached on E and F are compatible with their corresponding structures, it's easy to a bundle determines a preferred class of connections. We now examine to see that our connection on E ⊕ F is (G × H)-compatible. a variety of other situations in which a given vector bundle may inherit a natural class of connections|or a unique preferred connection|from some Example 4.1 (Metrics on direct sums). Let us see how the general external structure. A particularly important case is considered in x4.2 and framework just described applies to the previously mentioned construction x4.3, where E is the tangent bundle of M, and we find that there is a of a bundle metric on E ⊕F when E and F are both Euclidean bundles. In preferred metric connection for any Euclidean structure on T M. This is this case, the structure groups of E and F are O(m) and O(`) respectively, the fundamental fact underlying Riemannian geometry. so the direct sum has structure group O(m) × O(`), which is included 95 4.1. SUMS, PRODUCTS, LINEAR MAPS 97 98 CHAPTER 4. NATURAL CONSTRUCTIONS naturally in O(m+`) by (4.1). Thus E ⊕F inherits an O(m+`)-structure, Applying Exercise 4.2 to the bilinear pairing Hom(E; F ) ⊕ E ! F : that is, a bundle metric. To see that it matches the metric we defined (A; v) 7! Av, we find that the covariant derivatives on these three bun- earlier, we only need observe that by definition, any pair of orthogonal dles are related by frames for Ex and Fx defines an O(m)×O(`)-compatible frame for Ex ⊕Fx, which is therefore also an orthogonal frame. rX (Av) = (rX A)v + A(rX v) (4.2) for A 2 Γ(Hom(E; F )) and v 2 Γ(E). In particular for the dual bundle Tensor products E∗ = Hom(E; M ×F), one chooses the trivial connection on M ×F (i.e. r = d) so that for α 2 Γ(E∗) and v 2 Γ(E), (4.2) becomes Things are slightly more interesting for the tensor product bundle E ⊗ F . Parallel transport is defined naturally by the condition LX (α(v)) = (rX α)(v) + α(rX v): (4.3) P t v w P t v P t w : γ( ⊗ ) = γ ( ) ⊗ γ( ) Observe that the left hand side of this expression has no dependence on the connection|evidently this dependence for E and E∗ on the right hand Indeed, since every element of E ⊗ F is a sum of such products, this t side cancels out. defines Pγ on E ⊗ F uniquely via linearity. Computation of the covariant derivative makes use of the following general fact: bilinear operations give rise to product rules. Tensor bundles and contractions Exercise 4.2. Suppose V , W and X are finite dimensional vector spaces, By the above constructions, a choice of any connection on a vector bundle β : V × W ! X is a bilinear map and v(t) 2 V , w(t) 2 W are smooth E ! M induces natural connections on the tensor bundles paths. Then k ` ∗ k E` = ⊗ E ⊗ ⊗ E : d β(v(t); w(t)) = β(v_(t); w(t)) + β(v(t); w_ (t)): k dt Recall that one can interpret the fibers (E` )p as spaces of multilinear maps ∗ ∗ F It follows that the covariant derivative satisfies a Leibnitz rule for the Ep × : : : × Ep × Ep × : : : × Ep ! ; tensor product: if γ_ (0) = X 2 TxM, ` k | {z } | {z } 0 d −1 and by convention E is the trivial line bundle M × F; for this it is natural r (v ⊗ w) = P t [v(γ(t)) ⊗ w(γ(t))] 0 X dt γ to choose the trivial connection r = d. Then the convariant derivative t=0 d satisfies a Leibnitz rule on the tensor algebra: = (P t)−1(v(γ(t))) ⊗ (P t)−1( w(γ(t))) dt γ γ t=0 rX (S ⊗ T ) = rX S ⊗ T + S ⊗ rX T: = rX v ⊗ w + v ⊗ rX w: We express this more succinctly with the formula It also commutes with contractions: recall that there is a unique linear bundle map 1 0 r(v ⊗ w) = rv ⊗ w + v ⊗ rw: tr : E1 ! E0 with the property that tr(α ⊗ v) = α(v) for any v 2 E and α in the ∗ 1 i Bundles of linear maps corresponding fiber of E . Writing A 2 E1 in components A j with respect i to a chosen frame, tr(A) is literally the trace of the matrix with entries A j For the bundle Hom(E; F ) ! M, it is natural to define parallel transport (see Appendix A). Then combining the Leibnitz rule for the tensor product so that if A 2 Hom(Eγ(0); Fγ(0)) and v 2 Eγ(0), then with (4.3), we see that t t t Pγ (Av) = Pγ (A)Pγ(v): LX tr(A) = tr (rX A) 4.1. SUMS, PRODUCTS, LINEAR MAPS 99 100 CHAPTER 4. NATURAL CONSTRUCTIONS 1 for any A 2 Γ(E1 ). More generally for any p 2 1; : : : ; k + 1 and q 2 Proof. It follows from the various Leibnitz rules that (ii) and (iii) are equiv- k+1 k 1; : : : ; ` + 1, one can define a contraction operation tr : E`+1 ! E` by the alent. We can show the equivalence of (i) and (iii) by a direct computation: condition assume first that r is G-compatible, and choose a path γ(t) 2 M with γ(0) = p, γ_ (0) = X 2 TpM. Then 1 `+1 tr(α ⊗ : : : ⊗ α ⊗ v1 ⊗ : : : ⊗ vk+1) = αq(v ) · α1 ⊗ : : : ⊗ αq ⊗ : : : ⊗ α`+1 ⊗ v ⊗ : : : ⊗ v ⊗ : : : ⊗ v ; d p 1 p k+1 LX (T (v1; : : : ; vk)) = T (v1(γ(t)); : : : ; vk(γ(t))) dt t=0 where the hat notation is usedc to indicate the lack of the corresponding b d − − term. A similar argument then shows that for any such operation, = T (P t) 1(v (γ(t))); : : : ; ( P t) 1(v (γ(t))) dt γ 1 γ k t=0 r tr(T ) = tr (r T ) (4.4) X X d t −1 = T (Pγ ) (v1(γ(t))) ; : : : ; vk(p) + : : : k+1 dt t=0 for all T 2 Γ(E`+1 ). d t −1 Exercise 4.3. Verify (4.4) + T v (p); : : : ; (P ) (vk(γ(t))) 1 dt γ t=0 = T (r v ; : : : ; v (p)) + : : : + T (v (p); : : : ; r v ) ; Another perspective on compatibility X 1 k 1 X k If E ! M is a vector bundle with structure group G, we have thus far where we've used the obvious generalization of Exercise 4.2 for multilinear defined G-compatibility for a connection purely in terms of parallel trans- maps. Conversely if (iii) holds, then it follows from this calculation that port, which is not always the most convenient description. We shall now for any v1; : : : ; vk 2 Ep, see, at least in certain important special cases, how this definition can be d d framed in terms of covariant derivatives.
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