《黎曼几何》课程教学资源（英文讲义）Lec13 GEODESICS ON RIEMANNIAN MANIFOLDS

LECTURE13:GEODESICSONRIEMANNIANMANIFOLDSAfter defining geodesics as “self-parallel curves" on any smooth manifold withlinear connection, today we will put the Riemannian metric structure into this pic-ture and study what do we gain with this new structure (for the geodesics as self-parallel curves and as integral curves, for the exponential map, and for the normalcoordinates etc).1.GEODESICSAS INTEGRALCURVESI "Speed" of a geodesics.Let (M,g) be a Riemannian manifold, and : [a, b] -→ M a smooth curve in M.Recall that is a geodesics if and only if it is self-parallel, i.e. V.= 0. By metriccompatibility,d(%, 1) = V4(%, ) = (仅4%,5) +(%, 4) = 0.As a result, we getProposition 1.1.If is a geodesic on a Riemannian manifold, then /il must be aconstant for all t.Note that this also implies that a re-parametrization of a geodesic is again ageodesic if and only if the re-parametrization is a linear re-parametrizationIn particular, on a Riemannian manifold one can always re-parameterize a geo-desic so thatits“speed"is l:Definition 1.2.We will call a geodesics on a Riemannian manifold satisfyingI(t) = 1 a normal geodesics.Of course given any geodesic, the corresponding normal geodesic is nothing elsebut the arc-length re-parametrization of the given geodesic.I Geodesics as integral curves at the presence of metric.Last time by introducing y' = r' we converted the system of second order ODEsfor a geodesic to a system of first order ODEs[ih =yh,1≤k≤mIgk=-rhiy'y,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 the1

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 pic￾ture and study what do we gain with this new structure (for the geodesics as self￾parallel 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 geo￾desic 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

2LECTURE13:GEODESICSONRIEMANNIANMANIFOLDSintegral curve of the vector fieldaoX=ykiiyOrkOykAlthough one can show that the vector field X 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 bun-dle, then there is a geometrically important vector field whose integral curves givegeodesics on M. Recall that given any coordinate chart (U, rl,..: , rm) on M, any1-form w can be expressed locally on U as w = Sidr and as a result, one gets acoordinate chart (T*U,rl,..,rm,Si,...,Sm)for the cotangent bundle T*M.Now given a Riemannian metric g on M, i.e. an inner product on each tangentspace, one gets a dual innerproduct on each cotangent space.Consider the smoothfunction defined on T*M / [O] by152qi(r)sEjf(r,s) = 2Definition 1.3.The Hamiltonian vector field of f isfaofaHIt is a vector field on T*M / [0} which preserves f (and thus preserves [5l),H(f) = 0.As a consequence, it defines a vectorfield on each level set of f,and in particularonthecospherebundleS*M = (r, E) I IIllr = 1).By definition the integral curves of H, are the curves T = T(t) such thatr(t) = H,(r(t).More precisely,if we denoteT(t) = (r'(t),..,rm(t),si(t),...,sm(t))then any integral curve of H satisfies the following Hamilton equations[=影,af[5=-]The flow generated by H, on S*M is called the geodesic flow of (M, g), whichis very important in studying Riemannian manifolds. Now we proveTheorem 1.4. Any integral curve of H, on S*M, when projected onto M, is anormal geodesic in M. Conversely, any normal geodesic in M arises in this way

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 bun￾dle, 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

3LECTURE13:GEODESICSONRIEMANNIANMANIFOLDSProof. Let I(t) = (r'(t), ...,rm(t),si(t), ...,Sm(t)) be an integral curve of HfthentheHamiltonequationsbecome-f_1qiSiSjneo=geaEkof10gjE20kS5OrkFrom the first equation we get Ek = guil. Put this into the second equation, we have10gjOgk + gul =2OrkJus'gnjinOriNote thatOg'jOgnl,OguiglignjOrkSnjOrkOrktheequationbecomesm=-0+10mtnogki18gm20rkOri20rkOriIn other words,Ogikri0gk+10g）1ogk0gitr)=0k2920rkOriOriOrkOriwhich is exactly the geodesic equation sincer',=295g(8;gki + 0igjk- 0kgi).So the projected curve (t) = (r'(t), ..: , rm(t)) is a geodesic on M. It is normalsinceghihil=ghghig"ssi=gje,si=1.Conversely, for any geodesic (t) = (r'(t),..., rm(t), we let Ek = gui. Thenthe above computations shows that r(t) = (r'(t),..., rm(t),si(t),...,Sm(t)) is an口integral curve of H, in S* M.Remark. The function s/? is the symbol of the Laplace-Beltrami operator Ag. Sothegeodesic flow is also closelyrelated to spectral geometry.Remark. As a consequence, (M,g) is geodesically complete if and only if the vectorfield 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 compactRiemannianmanifoldisgeodesicallycomplete

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

NLECTURE13:GEODESICSONRIEMANNIANMANIFOLDS2.THEEXPONENTIAL MAPAT THEPRESENCEOFMETRICI 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 exp,(Xp) is the end point of thegeodesic segment that starts at p in the direction of X, whose length equals [Xpl.In general the map exp, : &pnT,M -→ M is not a global diffeomorphism, even ifit may be defined everywhere in T,M. For example, on the round sphere Sm, exppis a diffeomorphism from any ball B,(o) C T,M of radius r< π to an open regionin Sm, but it fails to be injective on the ball B,(O) with r >T.Definition 2.1.The injectivity radius of Riemannian manifold (M,g) at p E M isinj,(M, g) := sup(r / exp, is a diffeomorphism on Br(O) c T,M),and the injectivity radius of (M, g) isinj(M,g) := inf[inj,(M,g) I p E M)Erample. inj(Sm, gsm) = T.Remark. If M is compact, then of course0 < inj(M, g) ≤ diam(M, g),where diam(M, g) = supp.geM d(p, q) is the diameter of (M,g). But for noncompactmanifolds M, we may have inj(M,g)= 0 or +oo.[But for any p, we always haveinj,(M,g) >0.]For any p < inj,(M, g), we have B,(O) C T,M n&, where B,(O) is the ball ofradius p in (T,M, gp) centered at 0.Definition 2.2. We will call B(p,p) = expp(Bp(0)) the geodesic ball of radius pcentered at p in M, and its boundary S(p,p) = B(p,p) the geodesic sphere ofradius p centered at p in M.Now let be any normal geodesic starting at p. Then for p < inj,(M,g), wehave (0, p) C B(p, p) and exp'((0, p) is the line segment in B,(0) c T,Mstarting at O in the direction whose length is p.As a consequence, the geodesicsstarting at p of lengths less than inj,(M, g) are exactly the images under exP, of linesegments starting at 0 of lengths no more than inj,(M, g). In particular,Corollary 2.3. Suppose p E M and p<inj,(M,g). Then for any q = expp(Xp) EB(p,p), the curve (t) =expp(tXp) is the unique normal geodesic connecting p to qwhose length is less than p.Remark.Nomatter how close p and g are to each other, one might be able to findother geodesics connecting p to q whose length is longer. To see this, one can lookat cylinders or torus, in which case one can always find infinitely many geodesicsconnecting two arbitrary given points p and q

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 to an open region in S m, but it fails to be injective on the ball Br(0) with 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 < inj(M, g) ≤ diam(M, g), where diam(M, g) = supp,q∈M d(p, q) is the diameter of (M, g). But for noncompact manifolds M, we may have inj(M, g) = 0 or +∞. [But for any p, we always have injp (M, g) > 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

LECTURE13:GEODESICSONRIEMANNIANMANIFOLDS5TGauss Lemma.Last time we showed that the exponential (dexpn)o=Id. Now let (p,X,) e&.By definition, exPp maps the point X, E TpM to the point expp(Xp) e M. Ingeneral, the differential dexp, at X, is no longer the identity map Id [In fact, if(dexpp)xp = Id for all p and Xp, then expp is an isometry from (T,M,gp) to (M,g) and thus(M,g) is flat.j. 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, X,) E &.Then for anyYp ET,M =Tx,(T,M), we have<(dexpp)x,Xp, (dexpp)x,Yp)exp(Xp) = (Xp, Yp)p.Proof. Without loss of generality, we may assume Xp,Yp + 0. By linearity, it'senough to check the lemma for Yp =X, and Yp I Xp.Case 1: Yp = Xp. If we denote (t) = exp(tX,), then X, =(0) and0expp(txp) =(1).(dexpp)x,Xp= Since geodesics are always of constant speed, we conclude<(d exPp)x,Xp, (d expp)x,Xp) = ((1), (1)) = ((0), (0)) = (Xp, Xp)Case 2: Yp I Xp. Under this condition one can find a curve i(s) in the sphere ofradius [X,l in T,M with (0) = Xp and %(0) = Yp. Since (p, Xp) E &, we see thatthere exists >0 so that for all 0<t <l and -e< s<e,(p, t(s)) e&.Let A={(t,s)/ o<t<1,-e<s<) and consider the smooth mapf : A → M, (t,s) → f(t,s) :=expp(ti(s)As usual we denote ft = df() and fs= df(). The by definition-dft(1, 0) =exp,(tX,)= (dexpp)x,X,dtt=df.(1,0) = =0exPp(i(s) = (dexpp)x,Ypand thus((dexpp)x,Xp, (dexpp)x,Yp) = (ft(1, 0), fs(1, 0))On the other hand, we have.for each fixed so, f(t, so)is a geodesic with tangent vector field ft. SoVuft=0.. since V is torsion free, Vf.ft -Vf.f = [fs, fl] = df([0s, ot]) = O and thusVfaft=Vffs

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