LECTURE 12: GEODESICS AS SELF-PARALLEL CURVES (ON MANIFOLDS WITH CONNECTION) Now we turn to the next topic in this course: geodesic, which is a generalization of the notion of straight line in the Euclidean space. As we know, a line in R m is both a curve “with constant direction”, and a curve that “minimize distances between any two points on it”. As a result, we will have two ways to define geodesics on Riemannian manifolds, which, as we will see, are equivalent. On the other hand, for the first method (i.e. regard geodesics as curves “with constant directions”), what we need is the existence of a covariant derivative instead of a Riemannian metric structure, and as a result, it works for any smooth manifold with a linear connection. So today we will introduce the first method, i.e., focus on “non-metric properties” of geodesics. 1. Geodesics on manifolds with linear connections ¶ Geodesics for manifolds with linear connections. Let M be a smooth manifold. To define a geodesic as a “curve with constant direction”, what we need is a structure that can be used to compare tangent vectors at different points along a curve, i.e. a parallel transport, or equivalently, a linear connection. So we let ∇ be a linear connection on M. Now suppose γ : [a, b] → M is a smooth curve in M. Then “γ is a geodesic” means that the tangent vector field γ˙ is “unchanged” along γ(under parallel transport), i.e. is covariantly constant along γ: Definition 1.1. We say γ is a geodesic if ˙γ is parallel along γ, i.e. ∇γ˙ (t)γ˙ = 0, ∀t. In local coordinates, if we write γ(t) = (x 1 (t), · · · , xm(t)), then γ˙(t) = dγ( d dt) = ˙x i (t)∂i . Now suppose X = Xi∂i is a smooth vector field near γ [If X is only defined on γ, then we need to extend it to a smooth vector field in a neighborhood of γ. By locality of ∇, the extension will not affect the computation below]. If we denote f i (t) = Xi (γ(t)), then ∇γ˙ (t)X i = ˙γ(t)X i = d dt(X i ◦ γ) = ˙f i (t) [i.e. the covariant derivative of any function along γ is its t-derivative] and thus (∇γ˙ X)|γ(t) = ( ˙γ(t)X i )∂i + Γk ijx˙ i (t)f j (t)∂k = ˙f k (t)∂k + Γk ijx˙ i (t)f j (t)∂k. As a result, the condition ∇γ˙ X = 0, i.e. “X is parallel along γ” becomes ˙f k (t) + Γk ij (γ(t)) ˙x i (t)f j (t) = 0, ∀k. 1
2LECTURE 12: GEODESICS AS SELF-PARALLEL CURVES (ON MANIFOLDS WITH CONNECTION) Apply this to the vector field X = ˙γ, we see γ is a geodesic if and only if locally its coordinate functions satisfy the following system of second order ODEs (1) ¨x k (t) + ˙x i (t) ˙x j (t)Γk ij = 0, 1 ≤ k ≤ m. Remark. A natural question is: Question: is a re-parametrization of a geodesics still a geodesic? Suppose γ is a geodesic and ˙γ ̸= 0(otherwise γ is constant), and ˜γ(s) = γ(t(s)) is a regular re-parametrization of γ, then ∇γ˜˙ (s) γ˜˙(s) = ∇γ˜˙ (s) (t ′ (s) ˙γ(t(s))) = ˙γ(t(s)) + (t ′ (s))2∇γ˙ (t(s))γ˙(t(s)) = t ′′(s) ˙γ(t(s)). So ˜γ is also a geodesic if and only if t ′′(s) = 0, i.e. t(s) = as + b for some constants a and b. So the answer to the above question is: Answer: A re-parametrization of a geodesics is still a geodesic if and only if the re-parametrization is linear. ¶ Basic examples. Example. Let M = R m, equipped with standard linear connection ∇ such that ∇XY = X(Y j )∂j , or equivalently, Γk ij = 0. Let γ be any curve and X be a vector field. Then for X to be parallel along γ, we need ˙f k (t) = 0 for all k, i.e. if and only if Xi ’s are constants on γ [so X is a constant vector field in R m along γ in the usual sense]. In particular, the geodesic equations in R m above become x¨ k (t) = 0, 1 ≤ k ≤ m. The solution to the system are linear functions, i.e. x k (t) = akt + bk for some constants ak, bk. As a consequence, γ is a geodesic if and only if it is the straight line in the direction ⃗a = ⟨a1, · · · , am⟩ that passes the point (b1, · · · , bm). Example. Consider M = S m the m-sphere, equipped with the Levi-Civita connection. For any p ∈ S m, regarded as a unit vector p = ⃗u ∈ R m+1, and for any unit tangent vector ⃗w ∈ TpS m, we let γ(t) = (cost) ⃗u + (sin t) ⃗w. be the great circle in S m passing p in the direction of ⃗w. Since the Levi-Civita connection on S m is given by ∇XY = ∇XY + ⟨X, Y ⟩⃗n, where ∇ is the Levi-Civita connection for R m+1, i.e. with Γ k ij = 0. So ∇γ˙ γ˙ = ∇γ˙ γ˙ + ⟨γ, ˙ γ˙⟩⃗n = ¨γ + ⃗n. But at the point γ(t), one has ⃗n = γ(t), and ¨γ(t) = −γ(t). So we get ∇γ˙ γ˙ = 0. In other words, any great circle on S m is a geodesic. [By uniqueness below, up to linear re-parametrizations they are essentially the only geodesics on S m]
LECTURE 12: GEODESICS AS SELF-PARALLEL CURVES(ON MANIFOLDS WITH CONNECTION)3 ¶ The existence, uniqueness and smoothness. To find a geodesic is equivalent to solve the system of second order ODEs (1). By introducing y i = ˙x i , we may convert it to a system of first order ODEs (with more variables and more equations) ( x˙ k = y k , y˙ k = −Γ k ijy i y j , 1 ≤ k ≤ m. So suppose we want to find a geodesic with γ(t0) = p = (p 1 , · · · , pm) and ˙γ(t0) = Xp = Xi∂i ∈ TpM, then we need to solve the above system with initial condition x(t0) = (x 1 (t0), · · · , xm(t0)) = p, y(t0) = (y 1 (t0), · · · , ym(t0)) = Xp. According to the fundamental theorem for systems of first order ODEs, • Existence: For any t0 ∈ R and any (p, Xp) ∈ TM, there is an open interval I ∋ t0 and open set U ∋ (p, Xp) so that for any (q, Xq) ∈ U, the system has a smooth solution γq,Xq (t) in t ∈ I with initial condition x(t0) = q, y(t0) = Xq. • Smooth dependence: The solution above, viewed as a map Υ(t, q, Xq) = γq,Xq (t), is a smooth map from I × U to M. • Uniqueness: If (x1, y1) is a solution of the system on an interval I1 ∋ t0, (x2, y2) is a solution of the system on an interval I2 ∋ t0, both with the initial condition (p, Xp) at t0, then (x1, y1) = (x2, y2) on I1 ∩ I2. As a consequence, we conclude Theorem 1.2. For any p ∈ M and any Xp ∈ TpM, there exists an ε > 0 and a unique geodesic γ = γp,Xp defined for |t| < ε such that γ(0) = p and γ˙(0) = Xp. Moreover, the map γ(t; p, Xp) = γp,Xp (t) depends smoothly on (t, p, Xp). Note that by uniqueness, for any (p, Xp) ∈ TM, there is a maximal interval ✿✿✿✿✿✿✿✿✿ Jp,Xp ⊂ R on which a geodesic γ with γ(0) = p and ˙γ(0) = Xp exists. Note that by the “linear re-parametrization remark” above, Jp,tXp = 1 t Jp,Xp . If Jp,Xp = R for all (p, Xp) ∈ TM, then we say (M, ∇) is geodesically complete. Remark. The dependence of the maximal interval J on the initial data (p, Xp) is not continuous: for example, one can consider in the punctured plane R 2 − {(0, 0)}. Then the geodesic starting at (−1, 0) in the direction ⟨1, 0⟩ has maximal existence interval (−∞, 1), while the geodesic starting at (−1, 0) in any other direction has maximal existence interval R. It is not hard to see that if M is compact, then it must be geodesically complete. We will see later that for Riemannian manifolds, (M, g) is geodesically complete if and only if as a metric space, (M, dist) is complete
4LECTURE 12: GEODESICS AS SELF-PARALLEL CURVES (ON MANIFOLDS WITH CONNECTION) 2. The exponential map and normal coordinates ¶ The exponential map. Let M be a smooth manifold endowed with a linear connection ∇. Consider E = {(p, Xp) | γp,Xp (t) is defined on an interval containing [0, 1]}. [So by definition E = TM if and only if (M, g) is geodesically complete.] By existence and smoothness above, for any (p, Xp) ∈ TM there is ε0 > 0 and an open neighborhood U of (p, Xp) so that for any (q, Xq) ∈ U, the maximal existence interval Jq,Xq of γq,Xq contains the interval (−ε0, ε0). As a result, Jq,ε0Xq/2 ⊃ (−2, 2), So E contains a neighborhood of the zero section M in TM. Note that E ∩ TpM is always a star-like subset in TpM for any p. Definition 2.1. The exponential map is defined to be exp : E → M, (p, Xp) 7→ expp (Xp) := γp,Xp (1). Example. For (R m, g0), we can identify each TpR m with R m. Then expp (Xp) = p+Xp. Example. For (S 1 , dθ ⊗ dθ), we can identify TeS 1 with R 1 . Then expe (Xp) = e iXp . Remark. Let M = G be a Lie group, endowed with the Levi-Civita connection of the bi-invariant metric on G, then expe coincides with the exponential map exp : g → G in Lie theory. In particular, if G is a matrix Lie group, then expe (A) = I + A + A2 2! + · · · + Ak k! + · · · . The smoothness of Υ(t; p, Xp) implies that the exponential map is smooth. In particular, for each p ∈ M, the map expp : TpM ∩ E → M is smooth. By definition expp maps 0 ∈ TpM to p ∈ M. As in Lie theory we also have the following useful lemma: Lemma 2.2. For any p ∈ M, if we identify T0(TpM) with TpM, then (d expp )0 = Id|TpM : TpM → TpM. Proof. for any Xp ∈ T0(TpM) = TpM, (d expp )0(Xp) = d dt t=0 expp (tXp) = d dt t=0 γ(1; p, tXp) = d dt t=0 γ(t; p, Xp) = Xp. □ So by the inverse function theorem, we immediately get Corollary 2.3. For any p ∈ M, there exists a neighborhood V of 0 in TpM and a neighborhood U of p in M so that expp : V → U is a diffeomorphism
LECTURE 12: GEODESICS AS SELF-PARALLEL CURVES(ON MANIFOLDS WITH CONNECTION)5 ¶ Normal neighborhoods and normal coordinates. So for any p ∈ M, there exists a neighborhood U ⊂ M of p and a neighborhood Ve ⊂ TpM of 0 so that the exponential map expp : Ve → U is a diffeomorphism. By ✿✿✿✿✿✿ fixing ✿✿ a ✿✿✿✿✿ basis✿✿✿✿✿ {ei}✿✿✿ of✿✿✿✿✿ TpM, we may identify Ve with an open subset V of R m, and as a result, the triple (exp−1 p , U, V ) form a local chart of M near p. Definition 2.4. If Ve is star-like, then we call U a normal neighborhood of p, call the local chart (exp−1 p , U, V ) a normal chart on M, and call the coordinate system {U; x 1 , · · · , xm} a normal coordinate system centered at p. By definition, the normal coordinate system centered at p has the nice characterizing property that any geodesic starting at p is given in such coordinates by γ : x(t) = (tv1 , tv2 , · · · , tvm), where (v 1 , · · · , vm) is the direction of the geodesics. Moreover, we have Lemma 2.5. Let {U; x 1 , · · · , xm} be a normal coordinate system centered at p. Then for all ⃗v ∈ R m and all 1 ≤ k ≤ m, Γ k ij (p)v i v j = 0. [In particular, if the linear connection ∇ is torsion free, then Γ k ij (p) = 0 for all i, j, k.] Proof. Put the parametric equation x(t) = (tv1 , tv2 , · · · , tvm). of a geodesic into the geodesic equation, we get for 1 ≤ k ≤ m, 0 = ¨x k (t) + Γk ij (γ(t)) ˙x i (t) ˙x j (t) = Γk ij (γ(t))v i v j . Letting t = 0, we get Γk ij (p)v i v j = 0 for all ⃗v and for any 1 ≤ k ≤ m. □ ¶ Normal convex neighborhoods. We may go a lot further. Theorem 2.6 (Whitehead). For any smooth manifold M with a linear connection, any p has a neighborhood U such that U is a normal neighborhood for any q ∈ U. Let’s explain the meaning before we prove the theorem. For any q, q′ ∈ U, since U is a normal neighborhood of q, there is a vector Xq→q ′ ∈ TqM so that γq,q′(t) := expq (tXq→q ′) is a geodesic from q = γ(0) to q ′ = γ(1) that lies entirely in U. Such an open set is called a convex normal neighborhood of p. So Whitehead theorem claims that any p admits a normal convex neighborhood. As a consequence, we can prove Corollary 2.7. Any smooth manifold M admits a good covering. Proof. Endow with M a linear connection ∇. Then by Whitehead theorem, each p ∈ M admits a normal convex neighborhood Up. Because each normal convex neighborhood is contractible [since it is diffeomorphic to a star-like subset in a vector space], and because arbitrary intersection of normal convex neighborhoods is still a normal convex neighborhood, they form a good covering of M. □
6LECTURE 12: GEODESICS AS SELF-PARALLEL CURVES (ON MANIFOLDS WITH CONNECTION) ¶ Proof of Whitehead theorem. Proof. Step 1: There is a neighborhood U of p such that for any q ∈ U, there is a normal chart (exp−1 q , Uq, Vq) with Uq ⊃ U. Take a neighborhoods U1 of p ∈ M and a neighborhood Ue1 of (p, 0) ∈ TM over U1 [i.e. π(Ue1) = U1, where π : TM → M is the bundle projection] such that for each q ∈ U1, • Ue1 is fiberwise star-like, i.e. Vq = Ue1 ∩ TqM is star-like in TqM, • the exponential map expq : Vq → Uq is a diffeomorphism. Consider the map Ψ : Ue1 → M × M, (q, Xq) 7→ (q, expq (Xq)). The Jacobian of Ψ at (p, 0) is I 0 I I . So Ψ is a local diffeomorphism, i.e. it maps a smaller neighborhood U1 ⊂ Ue1 diffeomorphically onto a neighborhood of (p, p) in M × M. In particular, one may find a neighborhood U of p in M so that U × U ⊂ Ψ(U1). By construction, Ψ−1 (U × U) ∩ TqM ⊂ U1 ∩ TqM ⊂ Vq and thus U ⊂ expq (Ψ−1 (U × U) ∩ TqM) ⊂ Uq. Step 2: U can be chosen to be normal with respect to any q ∈ U. We fix a normal normal chart (φ, U0, V0) centered at p, with normal coordinates x 1 , · · · , xm, where for simplicity we denote φ = exp−1 p . Apply Lemma 2.5 and shrink U0 if necessary, we may assume that the matrix δij − P k Γ k ijx k is “positive” at each point in U0, i.e. such that δij − P k Γ k ijx k v i v j ≥ 0 for all ⃗v ∈ R m and all q ∈ U0. We may assume Uq we get in Step 1 are all inside U0. Now we endow TpM with any inner product, and shrink U we get in Step 1 so that φ(U) is a ball of radius δ. By Step 1, for any q, q′ ∈ U, there is a vector Xq→q ′ ∈ TqM with expq (Xq ′ q ) = q ′ . Since Vq is star-like, the curve γq,q′(t) := expq (tXq ′ q ) is a geodesic from q = γ(0) to q ′ = γ(1) that lies in Uq. It remains to prove that γq,q′(t)(0 ≤ t ≤ 1) lies in U. Since the geodesic γq,q′ lies in U0, we work on its parametric equations x i = x i (t). Consider the function f(t) = P i (x i (t))2 . Then ¨f(t) = 2X i ( ˙x k (t))2 + ¨x k (t)x k (t) = 2X k ( ˙x k (t))2 − Γ k ijx˙ i (t) ˙x j (t)x k (t) = 2 δij − X k Γ k ijx k (t) γ(t) x˙ i (t) ˙x j (t) ≥ 0. As a consequence, f is convex and thus f(t) ≤ max{f(0), f(1)} for 0 ≤ t ≤ 1. Since q, q′ ∈ U, we have f(0), f(1) ≤ δ 2 . So the geodesic γq,q′ is inside U. □