LECTURE 13: GEODESICS ON RIEMANNIAN MANIFOLDS After defining geodesics as “self-parallel curves” on any smooth manifold with linear connection, today we will put the Riemannian metric structure into this picture and study what do we gain with this new structure (for the geodesics as selfparallel curves and as integral curves, for the exponential map, and for the normal coordinates etc). 1. Geodesics as integral curves ¶ “Speed” of a geodesics. Let (M, g) be a Riemannian manifold, and γ : [a, b] → M a smooth curve in M. Recall that γ is a geodesics if and only if it is self-parallel, i.e. ∇γ˙ γ˙ = 0. By metric compatibility, d dt⟨γ, ˙ γ˙⟩ = ∇γ˙⟨γ, ˙ γ˙⟩ = ⟨∇γ˙ γ, ˙ γ˙⟩ + ⟨γ, ˙ ∇γ˙ γ˙⟩ = 0. As a result, we get Proposition 1.1. If γ is a geodesic on a Riemannian manifold, then |γ˙ | must be a constant for all t. Note that this also implies that a re-parametrization of a geodesic is again a geodesic if and only if the re-parametrization is a linear re-parametrization. In particular, on a Riemannian manifold one can always re-parameterize a geodesic so that its “speed” is 1: Definition 1.2. We will call a geodesics γ on a Riemannian manifold satisfying |γ˙(t)| = 1 a normal geodesics. Of course given any geodesic, the corresponding normal geodesic is nothing else but the arc-length re-parametrization of the given geodesic. ¶ Geodesics as integral curves at the presence of metric. Last time by introducing y i = ˙x i we converted the system of second order ODEs for a geodesic to a system of first order ODEs ( x˙ k = y k , y˙ k = −Γ k ijy i y j , 1 ≤ k ≤ m using which we get the existence, smooth dependence and uniqueness of geodesics. In other words, the problem of finding a local geodesic is equivalent to finding the 1
2 LECTURE 13: GEODESICS ON RIEMANNIAN MANIFOLDS integral curve of the vector field Xe = y k ∂ ∂xk − Γ k ijy i y j ∂ ∂yk . Although one can show that the vector field Xe defined above is really globally defined (i.e. independent of the choice of coordinates), its geometric meaning is not that obvious. It turns out that if one transfer from the tangent bundle to the cotangent bundle, then there is a geometrically important vector field whose integral curves give geodesics on M. Recall that given any coordinate chart (U, x1 , · · · , xm) on M, any 1-form ω can be expressed locally on U as ω = ξidxi and as a result, one gets a coordinate chart (T ∗U, x1 , · · · , xm, ξ1, · · · , ξm) for the cotangent bundle T ∗M. Now given a Riemannian metric g on M, i.e. an inner product on each tangent space, one gets a dual inner product on each cotangent space. Consider the smooth function defined on T ∗M \ {0} by f(x, ξ) = 1 2 |ξ| 2 x = 1 2 g ij (x)ξiξj . Definition 1.3. The Hamiltonian vector field of f is Hf = X ∂f ∂ξi ∂ ∂xi − ∂f ∂xi ∂ ∂ξi . It is a vector field on T ∗M \ {0} which preserves f (and thus preserves |ξ|x), Hf (f) = 0. As a consequence, it defines a vector field on each level set of f, and in particular on the cosphere bundle S ∗M = {(x, ξ) | ∥ξ∥x = 1}. By definition the integral curves of Hf are the curves Γ = Γ(t) such that Γ(˙ t) = Hf (Γ(t)). More precisely, if we denote Γ(t) = (x 1 (t), · · · , xm(t), ξ1(t), · · · , ξm(t)), then any integral curve of Hf satisfies the following Hamilton equations ( x˙ k = ∂f ∂ξk , ˙ξk = − ∂f ∂xk . The flow generated by Hf on S ∗M is called the geodesic flow of (M, g), which is very important in studying Riemannian manifolds. Now we prove Theorem 1.4. Any integral curve of Hf on S ∗M, when projected onto M, is a normal geodesic in M. Conversely, any normal geodesic in M arises in this way
LECTURE 13: GEODESICS ON RIEMANNIAN MANIFOLDS 3 Proof. Let Γ(t) = (x 1 (t), · · · , xm(t), ξ1(t), · · · , ξm(t)) be an integral curve of Hf , then the Hamilton equations become x˙ k = ∂f ∂ξk = 1 2 g ijδikξj + 1 2 g ij ξiδjk = g kj ξj ˙ξk = − ∂f ∂xk = − 1 2 ∂gij ∂xk ξiξj From the first equation we get ξk = glkx˙ l . Put this into the second equation, we have ∂glk ∂xi x˙ ix˙ l + glkx¨ l = − 1 2 ∂gij ∂xk glix˙ l gnjx˙ n . Note that − ∂gij ∂xk glignj = g ij ∂gli ∂xk gnj = ∂gnl ∂xk , the equation becomes glkx¨ l = − ∂glk ∂xi x˙ ix˙ l + 1 2 ∂gnl ∂xk x˙ lx˙ n = − ∂gjk ∂xi x˙ ix˙ j + 1 2 ∂gji ∂xk x˙ ix˙ j . In other words, x¨ l = g kl(− ∂gjk ∂xi x˙ ix˙ j + 1 2 ∂gji ∂xk x˙ ix˙ j ) = − 1 2 g kl( ∂gjk ∂xi x˙ ix˙ j + ∂gik ∂xj x˙ jx˙ i − ∂gji ∂xk x˙ ix˙ j ), which is exactly the geodesic equation since Γ l ij = 1 2 g kl(∂jgki + ∂igjk − ∂kgij ). So the projected curve γ(t) = (x 1 (t), · · · , xm(t)) is a geodesic on M. It is normal since gklx˙ kx˙ l = gklg kjg liξj ξi = g ij ξj ξi = 1. Conversely, for any geodesic γ(t) = (x 1 (t), · · · , xm(t)), we let ξk = glkx˙ l . Then the above computations shows that Γ(t) = (x 1 (t), · · · , xm(t), ξ1(t), · · · , ξm(t)) is an integral curve of Hf in S ∗M. □ Remark. The function |ξ| 2 is the symbol of the Laplace-Beltrami operator ∆g. So the geodesic flow is also closely related to spectral geometry. Remark. As a consequence, (M, g) is geodesically complete if and only if the vector field Hf on S ∗M is complete. Note that if M is compact, then S ∗M is compact, and thus any smooth vector field on S ∗M is complete. As a result, any compact Riemannian manifold is geodesically complete
4 LECTURE 13: GEODESICS ON RIEMANNIAN MANIFOLDS 2. The exponential map at the presence of metric ¶ The injectivity radius. Now let’s turn to the exponential map and figure out what do we gain with g. For a Riemannian manifold, by definition the point expp (Xp) is the end point of the geodesic segment that starts at p in the direction of Xp whose length equals |Xp|. In general the map expp : Ep ∩TpM → M is not a global diffeomorphism, even if it may be defined everywhere in TpM. For example, on the round sphere S m, expp is a diffeomorphism from any ball Br(0) ⊂ TpM of radius r π. Definition 2.1. The injectivity radius of Riemannian manifold (M, g) at p ∈ M is injp (M, g) := sup{r | expp is a diffeomorphism on Br(0) ⊂ TpM}, and the injectivity radius of (M, g) is inj(M, g) := inf{injp (M, g) | p ∈ M}. Example. inj(S m, gSm) = π. Remark. If M is compact, then of course 0 0.] For any ρ < injp (M, g), we have Bρ(0) ⊂ TpM ∩ E, where Bρ(0) is the ball of radius ρ in (TpM, gp) centered at 0. Definition 2.2. We will call B(p, ρ) = expp (Bρ(0)) the geodesic ball of radius ρ centered at p in M, and its boundary S(p, ρ) = ∂B(p, ρ) the geodesic sphere of radius ρ centered at p in M. Now let γ be any normal geodesic starting at p. Then for ρ < injp (M, g), we have γ((0, ρ)) ⊂ B(p, ρ) and exp−1 p (γ((0, ρ))) is the line segment in Bρ(0) ⊂ TpM starting at 0 in the direction ˙γ whose length is ρ. As a consequence, the geodesics starting at p of lengths less than injp (M, g) are exactly the images under expp of line segments starting at 0 of lengths no more than injp (M, g). In particular, Corollary 2.3. Suppose p ∈ M and ρ < injp (M, g). Then for any q = expp (Xp) ∈ B(p, ρ), the curve γ(t) = expp (tXp) is the unique normal geodesic connecting p to q whose length is less than ρ. Remark. No matter how close p and q are to each other, one might be able to find other geodesics connecting p to q whose length is longer. To see this, one can look at cylinders or torus, in which case one can always find infinitely many geodesics connecting two arbitrary given points p and q
LECTURE 13: GEODESICS ON RIEMANNIAN MANIFOLDS 5 ¶ Gauss Lemma. Last time we showed that the exponential (d expp )0 = Id. Now let (p, Xp) ∈ E. By definition, expp maps the point Xp ∈ TpM to the point expp (Xp) ∈ M. In general, the differential d expp at Xp is no longer the identity map Id [In fact, if (d expp )Xp = Id for all p and Xp, then expp is an isometry from (TpM, gp) to (M, g) and thus (M, g) is flat.]. However, we can prove that expp is always a “radial isometry”: Lemma 2.4 (Gauss lemma). Let (M, g) be a Riemannian manifold and (p, Xp) ∈ E. Then for any Yp ∈ TpM = TXp (TpM), we have ⟨(d expp )XpXp,(d expp )Xp Yp⟩expp (Xp) = ⟨Xp, Yp⟩p. Proof. Without loss of generality, we may assume Xp, Yp ̸= 0. By linearity, it’s enough to check the lemma for Yp = Xp and Yp ⊥ Xp. Case 1: Yp = Xp. If we denote γ(t) = exp(tXp), then Xp = ˙γ(0) and (d expp )XpXp = d dt t=1 expp (tXp) = ˙γ(1). Since geodesics are always of constant speed, we conclude ⟨(d expp )XpXp,(d expp )XpXp⟩ = ⟨γ˙(1), γ˙(1)⟩ = ⟨γ˙(0), γ˙(0)⟩ = ⟨Xp, Xp⟩. Case 2: Yp ⊥ Xp. Under this condition one can find a curve γ1(s) in the sphere of radius |Xp| in TpM with γ1(0) = Xp and ˙γ1(0) = Yp. Since (p, Xp) ∈ E, we see that there exists ε > 0 so that for all 0 < t < 1 and −ε < s < ε, (p, tγ1(s)) ∈ E. Let A = {(t, s) | 0 < t < 1, −ε < s < ε} and consider the smooth map f : A → M, (t, s) 7→ f(t, s) := expp (tγ1(s)). As usual we denote ft = df( d dt) and fs = df( d ds ). The by definition ft(1, 0) = d dt t=1 expp (tXp) = (d expp )XpXp, fs(1, 0) = d ds s=0 expp (γ1(s)) = (d expp )Xp Yp and thus ⟨(d expp )XpXp,(d expp )Xp Yp⟩ = ⟨ft(1, 0), fs(1, 0)⟩. On the other hand, we have • for each fixed s0, f(t, s0) is a geodesic with tangent vector field ft . So ∇ftft = 0. • since ∇ is torsion free, ∇fs ft − ∇ftfs = [fs, ft ] = df([∂s, ∂t ]) = 0 and thus ∇fs ft = ∇ftfs
6 LECTURE 13: GEODESICS ON RIEMANNIAN MANIFOLDS • Since γ1 lies in the sphere of radius |Xp|, the length |ft | = |γ1(s)| = |Xp| is a constant. As a consequence of these three facts, ∂ ∂t⟨fs, ft⟩ = ⟨∇ftfs, ft⟩ + ⟨fs, ∇ftft⟩ = ⟨∇fs ft , ft⟩ = 1 2 ∇fs ⟨ft , ft⟩ = 0, i.e. ⟨ft , fs⟩ is independent of t. Since lim h→0 fs(h, 0) = lim h→0 d ds s=0 expp (hγ1(s)) = limt→0 d(expp )hXp (hYp) = 0, we conclude ⟨ft(1, 0), fs(1, 0)⟩ = 0, which proves the lemma. □ Geometrically, Gauss lemma implies Corollary 2.5 (The Geometric Gauss Lemma). For any ρ < injp (M, g) and any q ∈ S(p, ρ), the shortest geodesic connecting p to q is orthogonal to S(p, ρ). ¶ Local shortest curves are geodesics. As a consequence of Gauss lemma, we may strengthen Corollary 2.3 to Theorem 2.6. Suppose p ∈ M and δ < injp (M, g). Then for any q = expp (Xp) ∈ B(p, δ), the geodesic γ(t) = expp (tXp)(0 ≤ t ≤ 1) is the only piecewise smooth curve connecting p and q with length d(p, q). Proof. Let σ : [0, 1] → M be any piecewise smooth curve with σ(0) = p, σ(1) = q, and parameterized with constant speed. We want to show L(σ) ≥ d(p, q), with equality holds if and only if σ = γ. Without loss of generality, we may assume p ̸∈ σ((0, 1]) [otherwise we may take t0 = sup{t|σ(t) = p} and consider the curve σ|[t0,1] instead] and assume σ((0, 1)) ⊂ B(p, δ) [otherwise we may take t1 = inf{t|σ(t) ∈ S(p, δ)} and consider the curve σ|[0,t1] instead]. As a result, there exits unit vectors w(t) ∈ SpM and real numbers r(t) ∈ (0, δ] such that σ(t) = expp (r(t)w(t)). It follows σ˙(t) = (d expp )r(t)w(t)(r ′ (t)w(t) + r(t) ˙w(t)). Note that w(t) ∈ SpM for all t implies w(t) ⊥ w˙(t). So by Gauss lemma, (d expp )r(t)w(t)(r ′ (t)w(t)) ⊥ (d expp )r(t)w(t)(r(t) ˙w(t)) and thus |σ˙(t)| 2 ≥ ⟨(d expp )r(t)w(t)(r ′ (t)w(t),(d expp )r(t)w(t)(r ′ (t)w(t)⟩ = |r ′ (t)| 2
LECTURE 13: GEODESICS ON RIEMANNIAN MANIFOLDS 7 So if we denote b = Length(σ), then b = |σ˙(t)| at all smooth points t of σ and thus b = Length(σ) = Z 1 0 |σ˙(t)|dt = 1 b Z 1 0 |σ˙(t)| 2 dt ≥ 1 b Z 1 0 |r ′ (t)| 2 dt ≥ 1 b Z 1 0 |r ′ (t)|dt2 ≥ 1 b Z 1 0 r ′ (t)dt2 ≥ δ 2 b , where we used Cauchy-Schwartz inequality and the fact r(1) ≤ δ. It follows that b ≥ δ as desired. Moreover, if the equality holds, then ˙w = 0 and |r ′ (t)| is constant, which implies that σ is precisely the geodesic γ(t) = expp (tXp). □ ¶ Riemannian metric tensor in Riemannian normal coordinate system. Now we turn to normal coordinate systems for Riemannian manifolds. Since the Levi-Civita connection is torsion-free, we have seen that with respect to any normal coordinate system centered at p, Γ k ij (p) = 0, 1 ≤ i, j, k ≤ m. So what do we gain from the metric? Recall that behind a normal coordinate system (exp−1 p , U, V ) there hides an identification between Ve = exp−1 p (U) ⊂ TpM and V ⊂ R m, which is realized after a choice of a basis ei of TpM. For a Riemannian manifold (M, g), we will always identify Ve = exp−1 p (U) ⊂ TpM and an open subset V ⊂ R m by ✿✿✿✿✿✿✿✿✿ choosing✿✿✿ an✿✿✿✿✿✿✿✿✿✿✿✿✿✿ orthonormal✿✿✿✿✿✿ basis✿✿✿✿✿✿✿✿✿✿✿✿✿✿ {e1, · · · , em} ✿✿ of✿✿✿✿✿✿ TpM, and call the resulting normal coordinate system a Riemannian normal coordinate system at p. With a Riemannian normal coordinate system at hand, we can prove the following stronger result[c.f. formula (10) in Lecture 6]: Lemma 2.7. Let (M, g) be a Riemannian manifold, and {U; x 1 , · · · , xm} be a Riemannian normal coordinate system centered at p. Then (1) For all 1 ≤ i, j ≤ m, gij (p) = δij . (2) For all 1 ≤ i, j, k ≤ m, ∂kgij (p) = 0. (3) G(p) = 1 and ∂iG(p) = 0 for all 1 ≤ i ≤ m, where G = det(gij ). Proof. (1) By definition of Riemannian normal coordinate system we have ∂i |p = d(expp )0ei = ei , which implies gij (p) = δij since {ei} is chosen to be orthonormal. (2) By metric compatibility we have ∂kgij (p) = ⟨∇∂k ∂i , ∂j ⟩(p) + ⟨∂i , ∇∂k ∂j ⟩(p) = Γl ki(p)glj (p) + Γl kj (p)gli(p) and thus the conclusion follows from the fact Γk ij (p) = 0. (3) This is a direct consequence of (1), (2) and the definition of determinant. □
8 LECTURE 13: GEODESICS ON RIEMANNIAN MANIFOLDS Remark. As a result, in a Riemannian normal coordinate centered at p, we have gij = δij + O(|x| 2 ) and det(gij ) = 1 + O(|x| 2 ) near p. In fact, as we will see later, what hides in O(x 2 ) are the curvature information of (M, g) at p: the Riemannian curvature for gij , and the Ricci curvature for det(gij ). In Riemannian normal coordinate system centered at p, many differential operators have very simple expressions at p. As a result, it can simplify computations a lot. For example, given any smooth vector field X = Xi∂i , we have defined its divergence to be divX = √ 1 G ∂i( √ GXi ). By Lemma 2.7 (3) we have ∂i( √ G)(p) = 0. So it follows that in a given Riemannian normal coordinate system centered at p, divX(p) = X i ∂iX i (p). As a result, the Laplacian ∆f at p also has a very simple expression, ∆f(p) = −div∇f(p) = −∂ 2 i f(p). Similarly the Hessian ∇2 f of f, in the Riemannian normal coordinates, becomes (∇2 f)(∂i , ∂j )(p) = ∂j∂if(p) − (∇∂j∂i)f(p) = ∂j∂if(p). In particular, we see that at each p, tr(∇2 f)(p) = g ij (p)(∇2 f)(∂i , ∂j )(p) = g ij (p)∂i∂jf(p) = ∂ 2 i f(p). So we proved Proposition 2.8. For any f ∈ C ∞(M), ∆f = −tr(∇2 f). This formula can be viewed as a second definition of the Laplace operator ∆. ¶ Strongly convex neighborhood. Finally we take a look at Whitehead’s theorem for Riemannian manifolds. We may carefully check the proof of Whitehead’s theorem last time: in step 2 we choose the convex normal neighborhood U carefully so that in the normal coordinate system, expp (U) is a ball in R m. In current setting if we use Riemannian normal coordinate system, then that means U is a small geodesic ball centered at p. Also in step 1 we may choose Ue1 carefully so that each Vq is a ball in (TqM, gq) instead of only a star-like subset in TqM, which means each Uq is a geodesic ball in the construction. In view of Theorem 2.6, we conclude that for such a geodesic ball U, any two points q1, q2 ∈ U can be connected by a unique geodesic γ of length d(q1, q2), and this minimizing geodesic γ lies in U Such a neighborhood is called strongly convex or geodesically convex. So we get Theorem 2.9 (Whitehead). Let (M, g) be a Riemannian manifold, then for any p ∈ M there exists ρ > 0 so that the geodesic ball B(p, ρ) is strongly convex