LECTURE12:GEODESICS AS SELF-PARALLEL CURVES(ON MANIFOLDS WITH CONNECTION)Now we turn to the next topic in this course: geodesic, which is a generalizationof the notion of straight line in the Euclidean space. As we know, a line in Rm is botha curve“with constant direction", and a curve that“"minimize distances betweenany two points on it".As a result, we will have two ways to define geodesics onRiemannian manifolds, which, as we will see, are equivalent. On the other hand, forthe first method (i.e. regard geodesics as curves "with constant directions"), what we need isthe existence of a covariant derivative instead of a Riemannianmetric structure, andas a result, it works for any smooth manifold with a linear connection. So today wewill introduce the first method, i.e., focus on"non-metric properties"of geodesics.1.GEODESICS ON MANIFOLDS WITHLINEAR CONNECTIONSI Geodesics for manifolds with linear connections.Let M be a smooth manifold.To define a geodesic as a “curve with constantdirection", what we need is a structure that can be used to compare tangent vectorsat different points along a curve, i.e. a parallel transport, or equivalently, a linearconnection.So we let be a linear connection on M.Now suppose:[a,b]→Mis a smooth curve in M. Then “"is a geodesic"means that the tangent vector fieldis“unchanged"along (under parallel transport), i.e. is covariantly constant along :Definition 1.1. We say is a geodesic if is parallel along , i.e.V(t)=0,Vt.In local coordinates, if we write (t) = (r'(t), ..., rm(t), then(t) = d(%) = (t)0):Now suppose X = Xio, is a smooth vector field near [If X is only defined on , then weneed to extend it to a smooth vector field in a neighborhood of .By locality of V,the extensionwill not affect the computation below). If we denote fi(t) =Xi((t)), thenV()Xi=(t)xi= %(X ) = f(t)dt[ie. the covariant derivative of any function along is its t-derivative] and thus(V,X)h(t) = (%(t)Xi)o, +Thij(t)f(t)O=j*(t)Ox+Fkg(t)f'(t)OkAs a result, the condition V,X =O, i.e.“X is parallel along "becomesjk(t)+rhi((t)a(t)fi(t) = 0, Vk
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
PECTURE12:GEODESICS AS SELF-PARALLEL CURVES (ON MANIFOLDS WITH CONNECTION)Apply this to the vector field X =, we seeis a geodesic if and only if locallyits coordinate functions satisfy the following system of second order ODEs(1)(t)+(t)(t)=0,1≤≤m.Remark. A natural question is:Question: is a re-parametrization of a geodesics still a geodesic?Suppose is a geodesic and O(otherwise is constant), and (s) = (t(s)) is aregular re-parametrization of , thenV()(s) = V(g)(t(s)(t(s)=(t(s)) + (t'(s))2V(t(s)(t(s)) = t"(s)(t(s)),So is also a geodesic if and only if t"(s) = o, i.e. t(s) = as + b for some constantsa and b. So the answer to the above question is:Answer: A re-parametrization of a geodesics is still a geodesic if andonly if the re-parametrization is linear.I Basic examples.Erample.Let M = Rm, equipped with standard linear connection such thatVxY = X(Yi)oj, or equivalently, Ihij = 0. Let be any curve and X be a vectorfield. Then for X to be parallel along , we need j*(t) = 0 for all k, i.e. if and onlyif Xi's are constants on [so X is a constant vector field in Rm along in the usual sense].In particular, the geodesic equations in Rm above become(t) =0,1<k<mThe solution to the system are linear functions, i.e.rk(t)=ast +bk for someconstants ak,bk. As a consequence, is a geodesic if and only if it is the straightline in the direction d = (ai,...,am) that passes the point (bi,.. ,bm).Erample.Consider M =Sm the m-sphere, equipped with theLevi-Civita connec-tion. For any p e sm, regarded as a unit vector p = i e Rm+i, and for any unittangentvectorweTm,welet(t) = (cost) ü+ (sint) w.be the great circle in sm passing p in the direction of w. Since the Levi-Civitaconnection on Sm is given byVxY=xY+(X,Y)n,whereis the Levi-Civitaconnection for Rm+1, i.e. with F'ij = 0. SoV=+,=+n.But at the point (t), one has n =(t), and (t) = -(t). So we getV=0.In other words, any great circle on Sm is a geodesic. [By uniqueness below, up to linearre-parametrizations they are essentially the only geodesics on Sm]
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:GEODESICSAS SELF-PARALLEL CURVES(ONMANIFOLDS WITHCONNECTION3I The existence, uniqueness and smoothness.To find a geodesic is equivalent to solve the system of second order ODEs (1)By introducing y=t, wemay convert it to a system of first order ODEs (withmore variables and more equations)[=y,1≤k≤m.(gk=-rkijy'y',So suppose we want to find a geodesic with (to) =p = (pl, ..,pm) and (to) =X, = X', E T,M, then we need to solve the above system with initial conditionr(to) = (r'(to),..-,rm(to)) = p, y(to) = (y'(to),..:,ym(to)) = Xp. According tothe fundamental theorem for systems of first order ODEs,. Existence: For any to E R and any (p,X,) eTM, there is an open intervalI to and open set u (p, X,) so that for any (q,X.) eu, the system has asmooth solution q,x (t) in t E I with initial condition r(to) = q, y(to) = Xq:. Smooth dependence: The solution above, viewed as a map r(t,q,X)=q,x,(t), is a smooth map from I x u to M.Uniqueness: If (ci,yi) is a solution of the system on an interval Ii to(c2,y2)is a solution of the system on an interval I2 to,both with the initialcondition (p,Xp)atto,then(r1,yi)=(r2,y2)on IinI2.As a consequence, we concludeTheorem 1.2. For any p E M and any X, E T,M, there erists an > 0 and aunique geodesic = p,x, defined for Itl < such that (0) = p and (O) = Xp.Moreover, the map (t; p, Xp) = p,x,(t) depends smoothly on (t,p, Xp).Note that by uniqueness, for any (p,Xp) E TM, there is a maximal intervalJp,X, C R on which a geodesic with (O) = p and (O) = X, exists. Note that bythe“linear re-parametrization remark"above,Jp,tXpp.XpIf Jp,x, = R for all (p, Xp) e TM, then we say (M, V) is geodesically complete.Remark. The dependence of the maximal interval J on the initial data (p,X,) isnot continuous: for example, one can consider in the punctured plane R?-f(o, O)).Then the geodesic starting at (-1, O) in the direction (1, ) has maximal existenceinterval (-oo, 1), while the geodesic starting at (-1, O) in any other direction hasmaximal 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 ifand only if as a metric space, (M, dist) is complete
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
LECTURE12:GEODESICS AS SELF-PARALLEL CURVES (ONMANIFOLDS WITH CONNECTION)2.THEEXPONENTIALMAPANDNORMALCOORDINATESI The exponential map.LetMbea smoothmanifold endowed witha linearconnection V.Consider& = [(p, X,) I p.x,(t) is defined on an interval containing [0, 1][So by definition & = TM if and only if (M,g) is geodesically complete.]By existence and smoothness above, for any (p, X,) e TM there is eo > 0 and anopen neighborhood u of (p, Xp) so that for any (q, Xq) eu, the maximal existenceinterval Ja.xg of q,X contains the interval (-Eo,eo). As a result,Jq,eoXa/2 (-2, 2),So & contains a neighborhood of the zero section M in TM. Note that &nT,M isalways a star-like subset in T,M for any p.Definition 2.1. The erponential map is defined to beexp : 8 → M, (p, Xp) → exPp(Xp) := p,x,(1).Erample.For (Rm,go), wecan identifyeachT,Rm with Rm.Then exPp(Xp)=p+Xp.Erample. For (S', do de), we can identify Ts1 with Ri. Then expe(X,) = eixp.Remark. Let M = G be a Lie group, endowed with the Levi-Civita connection of thebi-invariant metric on G, then expe coincides with the exponential map exp : g -→ Gin Lie theory. In particular, if G is a matrix Lie group, thenAkA2expe(A) = I +A +-.2!k!The smoothness of T(t; p, Xp) implies that the exponential map is smooth. Inparticular,foreach pEM,themapexPp:T,Mn&-Mis smooth. By definition exP, maps O e T,M to p e M. As in Lie theory we alsohave the following useful lemma:Lemma 2.2. For any p E M, if we identify To(T,M) with T,M, then(dexP,)o = IdlT,M : T,M -→ T,MProof. for any X, E To(T,M) = T,M,dddexpp(tXp) =(d expp)o(Xp) =(1;p,txp)=(t; p, Xp) = Xpdtat口Soby the inversefunction theorem, we immediately getCorollary 2.3. For any p E M, there erists a neighborhood V of 0 in TpM and aneighborhood U of p in M so that expp: V-U is a diffeomorphism
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
LECTURE12:GEODESICSASSELF-PARALLELCURVES(ONMANIFOLDSWITHCONNECTIONNormal neighborhoods and normal coordinates.So for any p e M, there exists a neighborhood U c M of p and a neighborhoodV c T,M of O so that the exponential map exp, : V→ U is a diffeomorphism. Byfixing a basis lel of TM, we may identify V with an open subset V of Rm, and asa result, the triple (exppl, U, V) form a local chart of M near p.Definition 2.4. If is star-like, then we call U a normal neighborhood of p, callthe local chart (exp,',U,V) a normal chart on M, and call the coordinate system[U; r', ... , r"] a normal coordinate system centered at p.By definition, the normal coordinate system centered at p has the nice charac-terizing property that any geodesic starting at p is given in such coordinates by: r(t) = (tul,tw2,...,tum),where (ul,...,um)is the direction of the geodesics.Moreover,we haveLemma 2.5.Let fU; rl,.., rmj be a normal coordinate system centered at p. Thenfor all u e Rm and all 1 ≤k ≤ m, Ikii(p)u'j= O. [In particular, if the linear connectionVistorsion free, then Tkig(p)=o forall i,j,k.)Proof. Put the parametric equation r(t) = (tul,tu?, ... ,tum). of a geodesic into thegeodesic equation, we get for 1≤k ≤m,0=(t) +((t)(t)(t) =((t)u.口Letting t=0,wegetIk(p)uwj=0 forall u and forany1≤k≤m.I 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 g E U.Let's explain themeaning before we prove the theorem.For any q,q eU, sinceU is a normal neighborhood of q, there is a vector Xq→q e TqM so thatq,q (t) := exPg(tXq→q))is a geodesic from q = (o) to q' = (1) that lies entirely in U. Such an open set iscalled a conver normal neighborhood of p. So Whitehead theorem claims that anyp admits a normal convex neighborhood. As a consequence, we can proveCorollary 2.7.Any smooth manifold M admits a good covering.Proof.Endow with M a linear connection V.Then by Whitehead theorem, eachp E M admits a normal convex neighborhood Up.Because each normal convexneighborhood is contractible [since it is diffeomorphic to a star-like subset in a vector space],and becausearbitraryintersectionof normal convexneighborhoodsis stillanormal口convex neighborhood, they form a good covering of M
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. □