LECTURE 15: COMPLETENESS 1. The Hopf-Rinow Theorem ¶ The Hopf-Rinow Theorem and consequences. Last time we proved that on any compact Riemannian manifold, any nontrivial path-homotopy class contains a shortest curve and that curve must be a geodesics. Today we will study geodesics in a wider class of Riemannian manifolds, namely, complete Riemannian manifolds, and prove the existence of shortest geodesics in each non-trivial path-homotopy class in such manifolds. We will first prove the existence of shortest geodesics [i.e. a geodesic of length d(p, q)] between any two given points p, q on any complete Riemannian manifold. This is the second part of a well-known theorem proved by Hopf and Rinow in 1931. Recall that a Riemannian manifold (M, g) is called ✿✿✿✿✿✿✿✿✿✿✿✿ geodesically✿✿✿✿✿✿✿✿✿✿ complete if the maximal defining interval of any geodesic on M is R. On the other hand, any Riemannian manifold (M, g) admits a Riemannian metric structure given by d(p, q) = inf{L(γ) | γ is a piecewise smooth curve connecting p to q}, and thus we can talk about the completeness of d: a metric space is ✿✿✿✿✿✿✿✿✿ complete if any Cauchy sequence in it converges. Now we state Hopf-Rinow theorem, which contains two parts: the first part claims that for Riemannian manifolds, the two notions of completeness coincide; while the second part claims the existences of shortest geodesic on such manifolds. Theorem 1.1 (Hopf-Rinow). Let (M, g) be a connected Riemannian manifold. (Part I) The following statements are equivalent: (1) (M, d) is a complete metric space. (2) (M, g) is geodesically complete. (3) There exists p ∈ M so that expp is defined for all Xp ∈ TpM. (4) [Heine-Borel property] Any bounded closed subset in M is compact. (Part II) Moreover, each of the previous statements implies (5) for any p, q ∈ M, there exists a geodesic of length d(p, q) connecting p and q. Definition 1.2. A connected Riemannian manifold (M, g) satisfying any of (1)-(4) is called a complete Riemannian manifold. Remark. Property (5) is NOT enough to guarantee that (M, g) is complete. For example, the open unit ball B1(0) in (R m, g0) satisfies (5), but is not complete. 1
2 LECTURE 15: COMPLETENESS Remark. For a general metric space, condition (1) does NOT imply condition (4): any infinite dimensional Banach or Hilbert space like l 2 is a counterexample. So as metric spaces, Riemannian manifolds are special (and nice) metric spaces. We list a couple immediate consequences of Hopf-Rinow theorem. Since any compact metric space is complete, we get another proof of Corollary 1.3. Any compact Riemannian manifold is geodesically complete. Since any two points can be connected by a geodesic, Corollary 1.4. If (M, g) is complete and connected, then for any p ∈ M, the exponential map expp : TpM → M is surjective. Since the Heine-Borel property is inherited by closed subsets, we have[warning: although any closed subspace of a complete metric space is complete, one cannot apply (1) here since “the induced metric on a submanifold S in the metric space (M, d)” is not the same as “the Riemannian distance generated by the induced Riemannian metric on S ⊂ (M, g)”] Corollary 1.5. Any closed submanifold of a complete Riemannian manifold, when endowed with the induced Riemannian metric, is complete. ¶ Proof of “Hopf-Rinow Theorem, Part II”. We first prove Part II of Hopf-Rinow theorem. More precisely, we prove (2)⇒(5) , or equivalently, its local version, namely (3)⇒(5 ′ ) , where (5 ′ ) for any q ∈ M, there exists a geodesic of length d(p, q) connecting p and q. Denote r = d(p, q). We have already seen that there exists 0 < δ < r so that the exponential map expp is a diffeomorphism from Bδ(0) ∈ TpM to B(p, δ) ∈ M. Note that the geodesic sphere S(p, δ) = expp (Sδ(0)) is compact. Since the distance function is continuous [c.f. lecture 3], there exists p0 ∈ S(p, δ) so that d(p0, q) = inf p ′∈S(p,δ) d(p ′ , q). Let γ be the normal geodesic from p to p0. By (3), γ is defined over R. Let A = {s ∈ [δ, r] | d(γ(s), q) = r − s}. We will show sup A = r, which implies γ(r) = q. To prove this, we first notice that δ ∈ A, since r = d(p, q) = inf p ′∈S(p,δ) (d(p, p′ ) + d(p ′ , q)) = δ + inf p ′∈S(p,δ) d(p ′ , q) = δ + d(γ(δ), q). So A is nonempty. Secondly, it’s easy to see that A is closed, since the function f(s) = d(γ(s), q) − r + s is continuous and A = f −1 (0) ∩ [δ, r]
LECTURE 15: COMPLETENESS 3 Now let s0 = sup A. Since A is nonempty and closed, s0 ∈ A. Suppose s0 0 so that (q, Yq) ∈ E for any q with d(q, p) < ε and any Yq ∈ TqM with |Yq| < 2ε. So if we take i large enough so that b − si < ε 2 and thus d(γ(si), p) < ε 2 , then γ(t; γ(si), εγ˙(si)) is defined for t ∈ [0, 1]. In other words, the geodesic γ1(t) = γ(t; γ(si), γ˙(si)) is well defined for 0 < t < ε. Since γ1 coincides with γ at si , they must be the same. In particular, γ can be defined for all t < si + ε 2 , which exceeds the upper bound b, a contradiction. Similarly by considering the “reverse geodesic” one also has a = −∞. So any normal geodesic on M, and thus any geodesic on M, has defining interval R
4 LECTURE 15: COMPLETENESS (2)⇒(3) This is obvious. ((M, g) is geodesically complete ⇔ E = TM. ) (3)+(5 ′ )⇒(4) Let K ⊂ M be a bounded closed set. Then there exists a constant C > 0 so that d(p, k) < C for all k ∈ K. According to (3) and (5 ′ ), K ⊂ expp (BC(0)), where BC(0) is the closed ball of radius C in TpM, which is compact in TpM. Since expp is smooth, expp (BC(0)) is also compact. Thus K, as a closed subset of a compact set, is compact. 2. Geodesics and Riemann covering map ¶ Lifting to the Riemannian covering. Next let’s turn to prove the existence of length minimizing geodesics in any path-homotopy class of curves connecting p to q. The idea is to straightforward: instead of working on piecewise smooth curves in M connecting given points p and q that lies in a given path-homotopy classes, we will move to the universal covering π : Mf → M of M and work on piecewise smooth curves starting with a fixed pe ∈ π −1 (p) and ends at the point qe ∈ π −1 (q) so that any curve connecting pe and qe projects to a curve in the given path-homotopy classes, and then we can apply the second part of Hopf-Rinow theorem. For the argument mentioned above to work, we need a couple ingredients. First, we need to lift the complete metric g on M to a complete metric on its universal covering Mf. Recall • Let M, N be connected smooth manifolds. A smooth map f : M → N is said to be a smooth covering map if (1) for any q ∈ N, there is a neighborhood V of q in N and open subsets Uα of M so that f −1 (V ) = ∪αUα, (2) for each α, f : Uα → V is a diffeomorphism, (3) these Uα’s are disjoint. As is well known, if f : M → N is a covering map, then – dim M = dim N and f is surjective, – fix any pα ∈ f −1 (q), any path (and path homotopy) starts at q in N admits a unique lifting to M that starts at pα, – moreover, if N is simply connected, then f is a global diffeomorphism. • If (M, gM) and (N, gN ) are Riemannian manifolds, then a smooth covering map π : M → N is called a Riemannian covering map if π ∗ gN = gM. Note: – given any smooth covering map π : M → N and any Riemannian metric on N, one may pullback that metric to M to make the covering map a Riemannian covering. [c.f. PSet 1 Problem 3] – any Riemannian covering map is a ✿✿✿✿✿ local✿✿✿✿✿✿✿✿✿✿ isometry. • We also need some standard properties of local isometries. Let f : (M, g) → (N, h) be a local isometry, then
LECTURE 15: COMPLETENESS 5 – M and N have “the same” Riemannian metrics at corresponding points, and thus the same Levi-Civita connection and the same Riemannian curvature at corresponding points [c.f. PSet 2 Problem 1]. – In particular, f maps geodesics into geodesics, and if f is a Riemannian covering, then the lifting of a geodesic is a geodesic. – for any piecewise smooth curve γ in M, one has |γ˙ |γ(t) = |dfγ(t)( ˙γ)|f(γ(t)) and thus L(γ) = L(f(γ)). Now we prove Proposition 2.1. Let (M, g) be a complete Riemannian manifold, and π : Mf → M be a smooth covering map. Then (M, π f ∗ g) is complete. Proof. For any pe∈ Mf and any ve ∈ TpeMf, we denote p = π(pe) and v = dπpe(ve). Then by definition of completeness, there is a geodesic γ : R → M with γ(0) = p and γ˙(0) = v. By the path-lifting property for covering space, there is a unique lifting γe : R → Mf with γe(0) = pe, which is a geodesic since it is the lifting of a geodesic. Moreover, since π : (M, π f ∗ g) → (M, g) is a local isometry and since π ◦ γe = γ, we get ˙γe(0) = (dπpe) −1 ( ˙γ(0)) = (dπpe) −1 (v) = v. e So the geodesic starts at pe in the direction ve is defined over R. □ ¶ Length minimizing curves in given path-homotopy class. As a consequence, we can extend Theorem 1.4 in Lecture 14 to complete Riemannian manifolds. Theorem 2.2. Let (M, g) be a complete connected Riemannian manifold, and p, q are two points in M. Then in each path-homotopy class of curves γ with γ(0) = p, γ(1) = q, there is a length-minimizing piecewise smooth curve and it is a geodesic. Proof. Consider the universal covering π : Mf → M. Equip Mf with the covering Riemannian metric π ∗ g. Given any path σ : [0, 1] → M connecting p and q in the given homotopy class, and given any pe∈ π −1 (p), there is a unique lifting σe : [0, 1] → Mf of σ with σe(0) = pe. Since (M, π f ∗ g) is complete, by Hopf-Rinow theorem, there is a minimizing geodesic γe from pe to qe := σe(1). Since π is a local isometry, the projection γ = π ◦ γe is a geodesic in M with γ(0) = p, γ(1) = q. Since Mf is simply connected, γe is path-homotopic to σe and thus γ is path-homotopic to σ. Finally suppose σ1 be any piecewise smooth curve in M from p to q in the given path homotopy class, then its lifting σe1 in Mf with starting point σe1(0) = pe must ends at qe, and thus by our choice of γe, L(γ) = Length(γe) ≤ Length(σe1) = L(σ1). So γ is the shortest curve in the given path homotopy class. □
6 LECTURE 15: COMPLETENESS ¶ The theorem of Ambrose. In Proposition 2.1 we start with a complete Riemannian manifold downstairs and a smooth covering map [topological information], and end with a complete Riemannian structure upstairs so that the map is a local isometry [geometric information]. It turns out that the theorem has an “inverse”, i.e. given an upstairs complete Riemannian manifolds and a local isometry f [geometric information], the downstairs metric must be complete and the map is a covering map [topological information]: Theorem 2.3 (Ambrose). Let (M, g) and (N, h) be connected Riemannian manifold, and f : (M, g) → (N, h) a local isometry. Suppose (M, g) is complete, then f is a smooth covering map, and (N, h) is complete. Note that “f : (M, g) → (N, h) a local isometry and (N, h) is complete” is not enough to guarantee f to be a covering map. We give an immediate consequence of Ambrose’s theorem, which will be used later in studying structures of Riemannian manifolds of non-positive sectional curvature: Corollary 2.4. Let (M, g) be a connected Riemannian manifold, and p ∈ M. If expp : TpM → M is a local diffeomorphism everywhere, then expp is a covering map. Proof. Note that the condition implies expp is defined on the whole TpM, and thus by Hopf-Rinow, (M, g) is complete. Endow TpM with the metric g¯ = (expp ) ∗ g, then • expp : (TpM, g¯) → (M, g) is a local isometry. • For any v ∈ TpM, the curve γ(t) = tv is a geodesic in (TpM, g¯) since its image expp (tv) is a geodesic in (M, g). In other words, exp0 : T0(TpM) → TpM is define on the whole T0(TpM), and thus by Hopf-Rinow, (TpM, g¯) is complete. So by Ambrose theorem, expp is a covering map. □ ¶ Proof of Ambrose’s theorem. We will prove this in four steps. Step 1 “Lift” geodesics in N to geodesics in M: Lemma 2.5. Under the conditions of the theorem, given any geodesic γ : [a, b] → N and any p ∈ f −1 (γ(a)), we can “lift” γ to a geodesic γ¯ : [a, b] → M so that γ(t) = f(¯γ(t)) and γ¯(a) = p. Moreover, the lift is unique. Proof. Since dfp : TpM → Tγ(a)N is a linear isometry, one can find a unique Xp ∈ TpM so that dfp(Xp) = ˙γ(a). Let γ¯ : [a, b] → M be the geodesic in M with (used completeness here) γ¯(a) = p, γ¯˙(a) = Xp. Then f ◦ γ¯ is a geodesic in N with same initial conditions as γ, and thus γ = f ◦ γ¯. The uniqueness of lift also follows directly from the fact that dfp : TpM → Tγ(a)N is a linear isometry. □
LECTURE 15: COMPLETENESS 7 Step 2 (N, h) is complete. Fix any point p ∈ N which lies in the image of f. For any geodesic γ starting at p, by Lemma 2.5, we can lift γ to a geodesic γ¯ starting at any p¯ ∈ f −1 (p). Since (M, g) is complete, γ¯ is a geodesic defined for all t. It follows that γ = f ◦ γ¯ is a geodesic defined for all t. So by Hopf-Rinow theorem, (N, ℏ) is complete. Step 3 f is surjective. Fix any point p ∈ N which lies in the image of f. For any q ∈ N, since (N, h) is complete, there is a minimizing geodesic γ from p to q. By Lemma 2.5, we can lift γ to a geodesic γ¯ starting at any p¯ ∈ f −1 (p). It follows that q ∈ f(¯γ) ⊂ Im(f). Step 4 Verify covering properties. Step 4.1 Construct V and Uα: Fix any q ∈ N, we may assume f −1 (q) = {pα}α∈I . Take δ small enough so that V = B(q, δ) is a normal geodesic ball. For each α, let Uα = B(pα, δ) ⊂ M. We note that each point in Uα can be connected to pα through a unique minimizing geodesic of length less than δ: if there exists p ′ ∈ Uα that can be connected to pα by two geodesics γ, γ′ of lengths less that δ starting at pα, then f(γ) and f(γ ′ ) are geodesic in N from q to f(p ′ ) of lengths less than δ, and thus we must have f(γ) = f(γ ′ ) =: ˜γ and thus γ˙(0) = (dfpα ) −1 (γ˜˙(0) = ˙γ ′ (0). Step 4.2 f −1 (V ) = ∪αUα: For any p ∈ f −1 (V ), let γ : [0, 1] → N be the minimal geodesic in V connecting f(p) to q, and γ¯ its lift starting at p. Then f(¯γ(1)) = γ(1) = q, so there exists α so that γ¯(1) = pα. Note L(¯γ) = L(γ) < δ, we conclude that p ∈ Uα. So f −1 (V ) ⊂ ∪αUα. Conversely, for any point p ∈ Uα, there is a minimal geodesic γ¯ : [0, 1] → M connecting pα to p with length < δ. It follows that γ = f ◦ γ¯ is a geodesic starting from q with length < δ. So f(p) = f(¯γ(1)) = γ(1) ∈ V, and thus Uα ⊂ f −1 (V ). Step 4.3 f : Uα → V is diffeomorphism: Since local isometry maps geodesics into geodesics, and geodesics are determined by initial values, we have expq [dfpα (Xα)] = f(exppα (Xα)) for any Xα ∈ TpαM. Moreover, when restricted to balls of radius δ, both expq and exppα are diffeomorphisms. Since df is an linear isomorphism which is also a diffeomorphism on the whole TpαM, we conclude that f = expq ◦dfpα ◦ exp−1 pα when restricted to balls of radius δ, so in particular f : Uα → V is a diffeomorphism. Step 4.4 For α ̸= β, Uα ∩ Uβ = ∅: Suppose there exists p ∈ Uα ∩ Uβ and α ̸= β. Let γ¯α and γ¯β be the minimal geodesic from p to pα and pβ respectively. Then f(¯γα) and f(¯γβ) are minimal geodesics in N, both from f(p) to q. It follows that f(¯γα) = f(¯γβ), and both γ¯α and γ¯β are lifts of f(¯γα) from p. By uniqueness of lift, γ¯α = ¯γβ and thus pα = pβ