Chapter 9 Parallel Transport and Geodesics Let (M; g) be a Riemannian manifold. From now on unless otherwise is said, we will denote r to be the Levi-Civita connection with respect to g. Furthermore, as r coincides with the covariant derivative in case when M is an Euclidean hypersurface and g is the first fundamental form, we will use also use the term covariant derivative for the Levi-Civita connection. 9.1. Parallel Transport 9.1.1. Parallel Transport Equation. On an Euclidean space, we can make good n n sense of translations of vectors as TpR is naturally identified with TqR for any other point q. However, on an abstract manifold M, the tangent spaces TpM and TqM may not be naturally related to each other, so it is non-trivial to make sense of translating a vector V 2 TpM to a vector in TqM. If one translates a vector V at p 2 Rn along a path γ(t) connecting p and q, then it is d ease to see that dt V (γ(t)) = 0 along the path. In other words, Dγ0(t)V = 0 where D is the directional derivative of Rn. Now on a Riemannian manifold we have the notion of the covariant derivative r. Using r in place of D in Rn, one can still define the notion of translations as follows: Definition 9.1 (Parallel Transport). Given a curve γ : [0;T ] ! M and a vector V0 2 Tγ(0)M, we define the parallel transport of V0 along γ(t) to be the unique solution V (t) to the ODE: rγ0(t)V (t) = 0 for any t 2 [0;T ] with the initial condition V (0) = V0. In terms of local coordinates F (u1; ··· ; un), the parallel transport equation can be expressed as follows. Let γ(t) = F (γ1(t); ··· ; γn(t)) and V (t) = V i(t) @ , then @ui dγi @ γ0(t) = dt @ui 225 226 9. Parallel Transport and Geodesics j @ rγ0(t)V (t) = r dγi @ V (t) dt @ui @uj i j dγ @V @ j k @ = + V Γij dt @ui @uj @uk j i dV @ j k dγ @ = + V Γij : dt @uj dt @uk Therefore, the parallel transport equation rγ0(t)V (t) = 0 is equivalent to dV k dγi + V jΓk = 0 for any k: dt ij dt Notably, the equation depends on γ0(t) only but not on γ(t) itself, so it also makes sense of parallel transporting V0 along a vector field X. It simply means parallel transport of 0 V0 along the integral curve γ(t) such that γ (t) = X(γ(t)). Even though the parallel transport equation is a first-order ODE whose existence and uniqueness of solutions are guaranteed, it is often impossible to solve it explicitly (and often not necessary to). Nonetheless, there are many remarkable uses of parallel n transports. When we have an orthonormal basis feigi=1 of TpM at a fixed point p 2 M, one can naturally extend it along a curve to become a moving orthonormal basis n fei(t)gi=1 along that curve. The reason is that parallel transport preserves the angle between vectors. Given a curve γ(t) on a Riemannian manifold (M; g), and let V (t) be the parallel transport of V0 2 TpM along γ, and W (t) be that of W0 2 TpM. Then one can check easily that d g V (t);W (t) = g r 0 V; W + g(V; r 0 W = 0 dt γ γ by the parallel transport equations rγ0 V = rγ0 W = 0. Therefore, we have g V (t);W (t) = g V0;W0 for any t. In particular, if V0 and W0 are orthogonal, then so are V (t) and W (t) for any t. If we take W0 = V0, by uniqueness theorem of ODE we have W (t) ≡ V (t). The above result shows the length of V (t), given by pg(V (t);V (t)), is also a constant. Hence, the parallel transport of an orthonormal basis remains to be orthonormal along the curve. 9.1.2. Holonomy and de Rham Splitting Theorem. Another remarkable conse- quence of the above observation is that parallel transport can be used to define an O(n)- action on TpM. Consider a closed piecewise smooth curve γ(t) where γ(0) = p 2 M. The curve needs not to be smooth at the closing point p. Given a vector V 2 TpM, we parallel transport it subsequentially along each smooth segment of γ. Precisely, suppose γ is defined on [t0; t1] [ [t1; t2] [···[ [tk−1; tk] with γ(t0) = γ(tk) = p and that γ is smooth on each (ti; ti1 ) and is continuous on [t0; tk]. We solve the parallel transport equation to get V (t) which is continuous on [t0; tk] and satisfies rγ0 V = 0 on each (ti; ti−1). We denote the final vector V (tk) by Pγ (V ) 2 TpM. It then defines a map Pγ : TpM ! TpM: By linearity of rγ0 , it is easy to show that Pγ is a linear map. Moreover, as Pγ (V ) and V have the same length, we have in fact Pγ 2 O(TpM), the orthogonal group acting on TpM. The set of all Pγ ’s, where γ is any closed piecewise smooth curve based at p, is in fact a group with multiplication given by compositions, and inverse given by parallel transporting vectors along the curve backward. We call this: 9.1. Parallel Transport 227 Definition 9.2 (Holonomy Group). The holonomy group of (M; g) based at p 2 M is given by: Holp(M; g) := fPγ : γ is a closed piecewise smooth curve on M based at pg: Exercise 9.1. Let (M; g) be a connected complete Riemannian manifold, and p and q be two distinct points on M. Show that Holp(M; g) and Holq(M; g) are related by conjugations (hence are isomorphic). When (M; g) is the flat Euclidean space Rn, parallel transport is simply translations. Any vector will end up being the same vector after transporting back to its based point. n Therefore, Holp(R ; δ) is the trivial group for any p. For the round sphere S2 in R3, we pick two points P , Q on the equator, and mark N to be the north pole. Construct a piecewise great circle path P ! N ! Q ! P , then one can see that what parallel transport along this path does is a rotation on vectors in 2 TP S by an angle depending on the distance between P and Q. For instance, when P and Q are antipodal, the parallel transport map is rotation by π. When P = Q, then the parallel transport map is simply the identity map. By varying the position of Q, one can obtain all possible angles from 0 to 2π. To explain this rigorously, one way is to solve the parallel transport equation in Definition 9.1. There is a more elegant to explain this after we learn about geodesics. It is remarkable that the holonomy group reveals a lot about the topological structure about a Riemannian manifold, and is a extremely useful tool for classification problems of manifolds. There is a famous theorem due to Ambrose-Singer that the Lie algebra of Holp(M; g) is related to how curved (M; g) is around p. Another beautiful theorem which demonstrates the usefulness of parallel transport is the following one due to de Rham. It is widely used in Ricci flow to classify the topology of certain class of manifolds by decomposing them into lower dimensional ones. Theorem 9.3 (de Rham Splitting Theorem - local version). Suppose the tangent bundle TM of a Riemannian manifold (M; g) can be decomposed orthogonally into TM = E1⊕E2 such that each of Ei’s is invariant under parallel transport, i.e. whenever V 2 Ei, any parallel transport of V stays in Ei. Then, M is locally a product manifold (N1; h) × (N2; k) ∗ ∗ such that TNi = Ei, and g is a locally product metric π1 h + π2 k. Proof. The proof begins by constructing a local coordinate system fxi; yαg such that @ @ span = E1 and span = E2. This is done by Frobenius’ Theorem which @xi @yα asserts that such that such local coordinate system would exist if each Ei is closed under the Lie brackets, i.e. whenever X; Y 2 Ei, we have [X; Y ] 2 Ei. Let’s prove E1 is closed under the Lie brackets (and the proof for E2 is exactly the same). The key idea is to use (8.7) that: [X; Y ] = rX Y − rY X: First pick an orthonormal basis fei; eαg at a fixed point p 2 M, such that span ei = E1 and span eα = E2. Extend them locally around p using parallel transport, then one would have rei = reα = 0 for any i and α. Now suppose X; Y 2 E1, we want to show rX Y 2 E1. Since g(Y; eα) = 0 and rX eα = 0, we have by (8.8): 0 = X g(Y; eα) = g(rX Y; eα): This shows rX Y ? eα for any α, so rX Y 2 E1. The same argument shows rY X 2 E1, and so does [X; Y ]. 228 9. Parallel Transport and Geodesics Frobenius’ Theorem asserts that there exists a local coordinate system fxi; yαg with @ @ span = E1 and span = E2. Next, we argue that the metric g is locally @xi @yα expressed as: i j α β g = gij dx ⊗ dx + gαβ dy ⊗ dy ; and that gij depends only on fxig, and gαβ depends only on fyαg.