LECTURE16:VARIATIONSOFLENGTHANDENERGYAlthough we defined geodesics as “self-parallel" curves, in the last several lec-tures we have seen that on Riemannian manifolds, geodesics are closely related to"lengthminimizing"curves:. (Lecture 13) on any Riemannian manifold, in a small neighborhood of anypoint, geodesics are precisely the shortest curves connecting endpoints.(Lecture 14and 15) on any completeRiemannian manifold, in eachpath-homotopy class, there exists a length minimizing curve and it is a geodesic.On the other hand, we also know the existence of geodesics which are not length min-imizing in the given path homotopy class [e.g. closed geodesics on Sm]. In what followswe take a closer look at the relation between geodesics and the length functional.1.GEODESICS AS CRITICALPOINTS OFENERGYFUNCTIONALI The Euler-Lagrange equation.For any p,q e M, considerCp = [: [a, b] → M I is piecewise smooth and (a) = p, (b) = q),One may ask: what property distinguish geodesics in Cpg from other curves? Oneof the answers should be“length-minimizing",at least locally.Now let's attack thisproblem by studying the length functional directly.Recall that the length of a piecewise smooth curve : [a, b] → (M, g) isL() = Length() =Ti(t)]dt.To find the minimum of such a functional, for simplicity let's first assume thatis inside a coordinate patch, and thus is given by a vector-valued function r(t)('(t), :.: , rm(t)). Consider a very general question in variational analysis:Given a smooth function f =f(t,r,i), find all the minimizer of thefunctionalI(r) = / f(t,r(t),i(t)dtin the set of all smooth curves r(t) = (r'(t),-.: ,rm(t) with fixedendpoints (a) = p,r(b) = q.Since this“space of smooth curves”is huge (namely, of"infinitely dimensional"), onecannot apply usual methods in calculus to find the minimizer. Fortunately, there is anew branch of mathematics called yariational calculus that is invented to handle such1
LECTURE 16: VARIATIONS OF LENGTH AND ENERGY Although we defined geodesics as “self-parallel” curves, in the last several lectures we have seen that on Riemannian manifolds, geodesics are closely related to “length minimizing” curves: • (Lecture 13) on any Riemannian manifold, in a small neighborhood of any point, geodesics are precisely the shortest curves connecting endpoints. • (Lecture 14 and 15) on any complete Riemannian manifold, in each pathhomotopy class, there exists a length minimizing curve and it is a geodesic. On the other hand, we also know the existence of geodesics which are not length minimizing in the given path homotopy class [e.g. closed geodesics on S m]. In what follows we take a closer look at the relation between geodesics and the length functional. 1. Geodesics as critical points of energy functional ¶ The Euler-Lagrange equation. For any p, q ∈ M, consider Cpq = {γ : [a, b] → M | γ is piecewise smooth and γ(a) = p, γ(b) = q}. One may ask: what property distinguish geodesics in Cpq from other curves? One of the answers should be “length-minimizing”, at least locally. Now let’s attack this problem by studying the length functional directly. Recall that the length of a piecewise smooth curve γ : [a, b] → (M, g) is L(γ) = Length(γ) = Z b a |γ˙(t)|dt. To find the minimum of such a functional, for simplicity let’s first assume that γ is inside a coordinate patch, and thus is given by a vector-valued function x(t) = (x 1 (t), · · · , xm(t)). Consider a very general question in variational analysis: Given a smooth function f = f(t, x, x˙), find all the minimizer of the functional I(x) = Z b a f(t, x(t), x˙(t))dt in the set of all smooth curves x(t) = (x 1 (t), · · · , xm(t)) with fixed endpoints x(a) = p, x(b) = q. Since this “space of smooth curves” is huge (namely, of “infinitely dimensional”), one cannot apply usual methods in calculus to find the minimizer. Fortunately, there is a new branch of mathematics called ✿✿✿✿✿✿✿✿✿✿✿ variational ✿✿✿✿✿✿✿✿ calculus that is invented to handle such 1
2LECTURE16:VARIATIONSOFLENGTHANDENERGYproblems.The idea is:convert the one“infinitely dimensional"problem [inwhich wehave infinitely many directions to move] to infinitely many “one-dimensional problems"[in which wefix one direction to move].Here is how it works in this example:Since weare studying the functional on curves with fixed endpoints, we may fix any smoothmap y(t) = (y'(t), ..-,ym(t)) with y(a) = y(b) = 0 and consider the correspondingone-parameter family of curves of the form r(t)+ey(t). So if r = r(t) is a minimizerofI,thenwemusthavedd0I(r+ey)f(t,r+ey,i+ey)dtdedeaf(t,r,i)yh+Qiafd ofykdt(t, r,i)(ork(t,r,a)-dtoikAs a result, we see that if r is a minimizer (or a critical point) of I, thenafd afork(t,z,2) (t,),1≤≤m,which is known as the Euler-Lagrange equation for the functional I.I Arc length v.s. energy.We may apply Euler-Lagrange equation above to thefunctionf(t, a(t),(t) = (((t), (t)a(t)1/2 = (gi(r(t)i(t)(t)/2.However, since there is a square root, the computation could be a bit messy. It turnoutthat thereisa small trickthat can simplifythecomputation a lot:instead ofthe length functional, we can work on the energy functional:E() =/h(t)Pdt.By the Cauchy-Schwartz inequality, for each piecewise smooth curve ,L(0)2 = ( h(t)at) ≤ (~ 12at) (["r(t)Pat) = 2(b- a)E(0),with equality holds if and only if li(t)l = constant. In particular we see thatalthough the length functional L() is independent of the choice of parametrizations,the energy functional does depend on the parametrizations (and on the length ofthe interval [a,bl): among different parametrizations of on fixed [a,b], E() isminimized on the “constant speed parametrization".As a consequence we can proveProposition 1.1. A curve : [a,b] → M in Cpq minimize the energy functionalE() if and only if it has constant speed and minimize the length functional L()
2 LECTURE 16: VARIATIONS OF LENGTH AND ENERGY problems. The idea is: convert the one “infinitely dimensional” problem [in which we have infinitely many directions to move] to infinitely many “one-dimensional problems” [in which we fix one direction to move]. Here is how it works in this example: Since we are studying the functional on curves with fixed endpoints, we may fix any smooth map y(t) = (y 1 (t), · · · , ym(t)) with y(a) = y(b) = 0 and consider the corresponding one-parameter family of curves of the form x(t)+εy(t). So if x = x(t) is a minimizer of I, then we must have 0 = d dε ε=0 I(x + εy) = d dε ε=0 Z b a f(t, x + εy, x˙ + εy˙)dt = Z b a ∂f ∂xk (t, x, x˙)y k + ∂f ∂x˙ k (t, x, x˙) ˙y k dt = Z b a ∂f ∂xk (t, x, x˙) − d dt ∂f ∂x˙ k (t, x, x˙) y k dt. As a result, we see that if x is a minimizer (or a critical point) of I, then ∂f ∂xk (t, x, x˙) = d dt ∂f ∂x˙ k (t, x, x˙), 1 ≤ k ≤ m, which is known as the Euler-Lagrange equation for the functional I. ¶ Arc length v.s. energy. We may apply Euler-Lagrange equation above to the function f(t, x(t), x˙(t)) = (⟨x˙(t), x˙(t)⟩x(t)) 1/2 = (gij (x(t)) ˙x i (t) ˙x j (t))1/2 . However, since there is a square root, the computation could be a bit messy. It turn out that there is a small trick that can simplify the computation a lot: instead of the length functional, we can work on the energy functional: E(γ) = 1 2 Z b a |γ˙(t)| 2 dt. By the Cauchy-Schwartz inequality, for each piecewise smooth curve γ, L(γ) 2 = Z b a |γ˙(t)|dt2 ≤ Z b a 1 2 dt Z b a |γ˙(t)| 2 dt = 2(b − a)E(γ), with equality holds if and only if |γ˙(t)| ≡ constant. In particular we see that although the length functional L(γ) is independent of the choice of parametrizations, the energy functional does depend on the parametrizations (and on the length of the interval [a, b]): among different parametrizations of γ on fixed [a, b], E(γ) is minimized on the “constant speed parametrization”. As a consequence we can prove Proposition 1.1. A curve γ : [a, b] → M in Cpq minimize the energy functional E(γ) if and only if it has constant speed and minimize the length functional L(γ)
3LECTURE16:VARIATIONSOFLENGTHANDENERGYProof. Suppose :[a,b] →M minimize E()but there exists E Cpq such thatL()<L(),then for the“constant speed re-parametrization":[a,bj-→M of,E() = L()?L()2(b-a))L()≤E(),2(b -a)2(b-a)which is a contradiction. So any minimizer of E() must also minimize L()Conversely, if : [a,b] -→ M has constant speed and minimize L(), but thereis another : [a,b] -→ M in Cpq with E() < E(), thenL() ≤ 2(b- a)E()< V2(b- a)E() = L()口a contradiction.Since any piecewise smooth curve can be reparametrized to have constant speed,to minimize L(), it is enough to minimize E()whose integrand is much simpler.Applying Euler-Lagrange equation tof(t,r(t),c(t)) = gij(r(t)i(t)ii(t)we get,for 1≤k≤m,=doki+2gkj (g2)= 2(gkji) +OrkdtdOrwhich, as we have seen in Lecture 13, is equivalent to the geodesic equation+rhi=0,1≤k≤mAmazingly enough, by this way we get not only all the geodesics that are lengthminimizing curves in Cpg and all the geodesics that are the length minimizing curvesin each path homotopy class of Cpq , but in fact we get ALL the geodesics connectingp and q:Theorem 1.2. For a Riemannian manifold (M,g), a curve : [a,b] -→ M in Cpais a geodesic if and only if it satisfies the Euler-Lagrange equation of the energyfunctional E().This gives another proof of the fact that any length minimizing curve is a ge-odesic, and also explains why there exist geodesics that are not length minimizingeven in the given path-homotopy class: those curves are only “critical points" ofEwhich need not be minimizing among all near-by curves.If one need to findthe minimizing geodesics, then as usual one canfurther calculate the second orderderivative le=oE(r+ey) in a coordinate system, using which one can show thatgeodesics are always length-minimizing locally in a neighborhood.1Although these curves are not length minimizing in Cpq, they are in fact length minimizingamong “nearby curves",namely among curves of the form r + ey in the computation above for esmall enough, since these curves are in the same path-homotopy class
LECTURE 16: VARIATIONS OF LENGTH AND ENERGY 3 Proof. Suppose γ : [a, b] → M minimize E(γ) but there exists γ ′ ∈ Cpq such that L(γ ′ ) < L(γ), then for the “constant speed re-parametrization” γe : [a, b] → M of γ ′ , E(γe) = 1 2(b − a) L(γe) 2 = 1 2(b − a) L(γ ′ ) 2 < 1 2(b − a) L(γ) 2 ≤ E(γ ′ ), which is a contradiction. So any minimizer of E(γ) must also minimize L(γ). Conversely, if γ : [a, b] → M has constant speed and minimize L(γ), but there is another γ ′ : [a, b] → M in Cpq with E(γ ′ ) < E(γ), then L(γ ′ ) ≤ p 2(b − a)E(γ ′ ) < p 2(b − a)E(γ) = L(γ), a contradiction. □ Since any piecewise smooth curve can be reparametrized to have constant speed, to minimize L(γ), it is enough to minimize E(γ) whose integrand is much simpler. Applying Euler-Lagrange equation to f(t, x(t), x˙(t)) = gij (x(t)) ˙x i (t) ˙x j (t) we get, for 1 ≤ k ≤ m, ∂gij ∂xk x˙ ix˙ j = d dt gkjx˙ j + d dt gikx˙ i = 2 ∂gkj ∂xi x˙ ix˙ j + 2gkjx¨ j , which, as we have seen in Lecture 13, is equivalent to the geodesic equation x¨ k + Γk ijx˙ ix˙ j = 0, 1 ≤ k ≤ m. Amazingly enough, by this way we get not only all the geodesics that are length minimizing curves in Cpq and all the geodesics that are the length minimizing curves in each path homotopy class of Cpq 1 , but in fact we get ALL the geodesics connecting p and q: Theorem 1.2. For a Riemannian manifold (M, g), a curve γ : [a, b] → M in Cpq is a geodesic if and only if it satisfies the Euler-Lagrange equation of the energy functional E(γ). This gives another proof of the fact that any length minimizing curve is a geodesic, and also explains why there exist geodesics that are not length minimizing even in the given path-homotopy class: those curves are only “critical points” of E which need not be minimizing among all near-by curves. If one need to find the minimizing geodesics, then as usual one can further calculate the second order derivative d 2 dε2 |ε=0E(x + εy) in a coordinate system, using which one can show that geodesics are always length-minimizing locally in a neighborhood. 1Although these curves are not length minimizing in Cpq, they are in fact length minimizing among “nearby curves”, namely among curves of the form x + εy in the computation above for ε small enough, since these curves are in the same path-homotopy class
4LECTURE16:VARIATIONSOFLENGTHANDENERGY2.FORMULASFORTHEFIRSTANDSECONDVARIATIONSThe calculations above are thought-provoking but has the shortcoming thatthey are performed in a chart. In what follows we take a global way to calculate thefirst and second derivatives, and also study variations which could be more general(withoutfixingendpoints)ormorerestrictive (with geodesicvariation)TVariations.For simplicity we start with smooth variations of a smooth curve:Definition 2.1. Let : [a,b] -→ M be a smooth curve, and >0.(1) A smooth variation of is a smooth map f : [a,b] × (-e,e) -→ M so thatf(t, 0) =(t)forall t e [a,bl.In whatfollows, wewill also denote (t)=f(t,s)(2) A variation f is called proper if for every s e (-e,e),%(a) =(a) and %(b) =(b).(3) A variation is called a geodesic variation if each is a geodesic.For simplicity we denote R = [a,b] × (-e,e). Let f : R -→ M be a smoothvariation of . Then E =f*TM is a vector bundle over R, on which we havean induced linear connection =f* (where is the Levi-Civita connection on(M, g). To be rigorous, in what follows we will calculate via V, and refer to theappendix of this section for the definition and properties of V.The variation f gives rise to two natural sections of E, namelyf(t,s) := (df)s(%)) e Tr(t,s)M = Et,sandf(t,s) := (df)ts(%)) ETr(t,s)M=Ets,where and are the coordinate vector fields on R. Note that by definitionft(to, so) =o(to)We are mainly interested in the restriction of the sections fs and ft to s = O, whichare in fact “vector fields along ". Obviously we have (t) = ft(t, 0) = (df)t,o(α)Definition2.2.Wewill callV(t) := fs(t, 0) = (df)t,o(the variation field of the variation f.Note that if is an embedded curve and f is an embedding, then both (t)and V(t) can be regarded as vector fields on M along in a natural way, and thecomputations below can be carried out via instead of
4 LECTURE 16: VARIATIONS OF LENGTH AND ENERGY 2. Formulas for the first and second variations The calculations above are thought-provoking but has the shortcoming that they are performed in a chart. In what follows we take a global way to calculate the first and second derivatives, and also study variations which could be more general (without fixing endpoints) or more restrictive (with geodesic variation). ¶ Variations. For simplicity we start with smooth variations of a smooth curve: Definition 2.1. Let γ : [a, b] → M be a smooth curve, and ε > 0. (1) A smooth variation of γ is a smooth map f : [a, b] × (−ε, ε) → M so that f(t, 0) = γ(t) for all t ∈ [a, b]. In what follows, we will also denote γs(t) = f(t, s). (2) A variation f is called proper if for every s ∈ (−ε, ε), γs(a) = γ(a) and γs(b) = γ(b). (3) A variation is called a geodesic variation if each γs is a geodesic. For simplicity we denote R = [a, b] × (−ε, ε). Let f : R → M be a smooth variation of γ. Then E = f ∗TM is a vector bundle over R, on which we have an induced linear connection ∇e = f ∗∇ (where ∇ is the Levi-Civita connection on (M, g)). To be rigorous, in what follows we will calculate via ∇e , and refer to the appendix of this section for the definition and properties of ∇e . The variation f gives rise to two natural sections of E, namely, fs(t, s) := (df)t,s( ∂ ∂s) ∈ Tf(t,s)M = Et,s and ft(t, s) := (df)t,s( ∂ ∂t) ∈ Tf(t,s)M = Et,s, where ∂ ∂s and ∂ ∂t are the coordinate vector fields on R. Note that by definition, ft(t0, s0) = ˙γs0 (t0). We are mainly interested in the restriction of the sections fs and ft to s = 0, which are in fact “vector fields along γ”. Obviously we have ˙γ(t) = ft(t, 0) = (df)t,0( ∂ ∂t). Definition 2.2. We will call V (t) := fs(t, 0) = (df)t,0( ∂ ∂s) the variation field of the variation f. Note that if γ is an embedded curve and f is an embedding, then both ˙γ(t) and V (t) can be regarded as vector fields on M along γ in a natural way, and the computations below can be carried out via ∇ instead of ∇e
LECTURE16:VARIATIONSOFLENGTHANDENERGY5I The first variation formula of energy for smooth variations.Now we compute the variation of E along given variation (without fixing endpoints):Let f(t, s) be a smooth variation of a smooth curve : [a, b] → M. By Proposition3.11 and Proposition 3.13, the derivative of E(s) is4E(%) =00[(Va/sft,ft)dt=2]。 %(%(t), %(0)dt = (Va/otfs, ft)dtdsApplying metric compatibility (i.e. Proposition 3.11) again, we get -m-ma(Va/atfs,ft)dt=/Sowe getTheorem 2.3 (The First Variation of Energy). Given any smooth variation f(t,s)ofa smoothcurve:[a,b]→M,dE(%) =(Va/αtfs fi)dt = (f.(t, s), fi(t, s)=b -(fs,Vajatft)dtdsIn particular,dE(%) = - (V(t), Vs()) dt - (V(a), (a) + (V(b),(b) dsIn particular, if f is a proper smooth variation, thenE(%) = -(V(t), V()) dtdsAgain we see that is a geodesic (i.e.V=O) if and only if is a criticalpoint of the energy functional E among all proper variations.I The first variation formula of length for smooth variations.Use the same way, one can calculate the first variation of the length. A trick tosimplify the computation is the following observation:o0=%f)-110(Va/afs,ft)=(Va/atfs%h%(t)/ =t）=LftThen following the same computation, one getsTheorem 2.4 (The First Variation of Length). Let f(t, s) be a smooth variation ofa smooth curve . Then(a)(6)L(%) = -(v(t),V(可dt-(v(a),(V(b),1(a)l1(6)11As an application we prove
LECTURE 16: VARIATIONS OF LENGTH AND ENERGY 5 ¶ The first variation formula of energy for smooth variations. Now we compute the variation of E along given variation (without fixing endpoints): Let f(t, s) be a smooth variation of a smooth curve γ : [a, b] → M. By Proposition 3.11 and Proposition 3.13, the derivative of E(γs) is d dsE(γs) = 1 2 Z b a ∂ ∂s⟨γ˙ s(t), γ˙ s(t)⟩dt = Z b a ⟨∇e ∂/∂sft , ft⟩dt = Z b a ⟨∇e ∂/∂tfs, ft⟩dt. Applying metric compatibility (i.e. Proposition 3.11) again, we get Z b a ⟨∇e ∂/∂tfs,ft⟩dt= Z b a ∂ ∂t⟨fs, ft⟩dt− Z b a ⟨fs,∇e ∂/∂tft⟩dt=⟨fs,ft⟩|t=b t=a− Z b a ⟨fs,∇e ∂/∂tft⟩dt. So we get Theorem 2.3 (The First Variation of Energy). Given any smooth variation f(t, s) of a smooth curve γ : [a, b] → M, d dsE(γs) = Z b a ⟨∇e ∂/∂tfs, ft⟩dt = ⟨fs(t, s), ft(t, s)⟩|t=b t=a − Z b a ⟨fs, ∇e ∂/∂tft⟩dt. In particular, d ds s=0 E(γs) = − Z b a V (t), ∇γ˙ (t)γ˙ dt − ⟨V (a), γ˙(a)⟩ + ⟨V (b), γ˙(b)⟩. In particular, if f is a proper smooth variation, then d ds s=0 E(γs) = − Z b a V (t), ∇γ˙ (t)γ˙ dt. Again we see that γ is a geodesic (i.e. ∇γ˙ γ˙ = 0) if and only if γ is a critical point of the energy functional E among all proper variations. ¶ The first variation formula of length for smooth variations. Use the same way, one can calculate the first variation of the length. A trick to simplify the computation is the following observation: ∂ ∂s|γ˙ s(t)| = ∂ ∂s⟨ft , ft⟩ 1 2 = 1 2 1 |ft | ∂ ∂s⟨ft , ft⟩ = 1 |ft | ⟨∇e ∂/∂tfs, ft⟩ = ⟨∇e ∂/∂tfs, ft |ft | ⟩. Then following the same computation, one gets Theorem 2.4 (The First Variation of Length). Let f(t, s) be a smooth variation of a smooth curve γ. Then d ds s=0 L(γs) = − Z b a V (t), ∇γ˙ (t) γ˙ |γ˙ | dt − V (a), γ˙(a) |γ˙(a)| + V (b), γ˙(b) |γ˙(b)| . As an application we prove
6LECTURE16:VARIATIONSOFLENGTHANDENERGYProposition 2.5.Let S be a closed submanifold of (M,g).Suppose is a geodesicfrom p S to q E S with L() = d(p, S). Then is perpendicular to S.Proof. For any y E TS. take a curve g : (-e.e) S with g(o) = g and o(o) = Let be a variation of with (O) = p and (l) = o(s), where l = L(). ThenV(a) = O and V(b) = u, and bu the first variation formula,d0=ls=0 L(%) = (u, (1),d口thus the conclusion follows.I Piecewise smooth curve.More generally, one can consider piecewise smooth curves : [a, bl → M, i.e.there exists a subdivisiona=to+V(b),ds15(6)12<v-)一The local computations above imply that among smooth curves, geodesics arecritical points of the energy functional.A natural question is:If a curve is piecewisesmooth, can it be a critical point of the energy functional? Of course for be acritical point of the energy functional, it mustbea geodesic when restricted toanysubinterval where it is smooth, or in other words, it must be"piecewise geodesic".Corollary 2.7. If a piecewise smooth curve is a critical point of the energy func-tional among proper variations, then it is Ci and thus a geodesic
6 LECTURE 16: VARIATIONS OF LENGTH AND ENERGY Proposition 2.5. Let S be a closed submanifold of (M, g). Suppose γ is a geodesic from p ̸∈ S to q ∈ S with L(γ) = d(p, S). Then γ is perpendicular to S. Proof. For any v ∈ TqS, take a curve σ : (−ε, ε) → S with σ(0) = q and ˙σ(0) = v. Let γs be a variation of γ with γs(0) = p and γs(l) = σ(s), where l = L(γ). Then V (a) = 0 and V (b) = v, and bu the first variation formula, 0 = d ds|s=0L(γs) = ⟨v, γ˙(l)⟩, thus the conclusion follows. □ ¶ Piecewise smooth curve. More generally, one can consider piecewise smooth curves γ : [a, b] → M, i.e. there exists a subdivision a = t0 < t1 < t2 < · · · < tk < tk+1 = b such that γ is smooth on each interval [ti , ti+1]. We shall consider piecewise smooth variations of γ, which is a continuous map f : [a, b] × (−ε, ε) → M so that f is smooth on each [ti , ti+1] × (−ε, ε) for each i. Note that this implies • for each s ∈ (−ε, ε), the curve t 7→ γs(t) = f(t, s) is piecewise smooth, • for each t ∈ [a, b], the curve s 7→ f(t, s) is smooth (so fs is well-defined at ti ’s ). Applying the previous theorems to each [ti , ti+1] × (−ε, ε), we get Corollary 2.6. Let f be a piecewise smooth variation of curve γ. Then d ds s=0 E(γs)=− Z b a ⟨V (t),∇γ˙ γ˙⟩ dt−⟨V (a),γ˙(a)⟩+⟨V (b),γ˙(b)⟩−X k i=1 V (ti),γ˙(t + i )−γ˙(t − i ) and d ds s=0 L(γs) = − Z b a V (t), ∇γ˙ γ˙(t) |γ˙ | dt − V (a), γ˙(a) |γ˙(a)| + V (b), γ˙(b) |γ˙(b)| − X k i=1 V (ti), γ˙(t + i ) |γ˙(t + i )| − γ˙(t − i ) |γ˙(t − i )| . The local computations above imply that among smooth curves, geodesics are critical points of the energy functional. A natural question is: If a curve is piecewise smooth, can it be a critical point of the energy functional? Of course for γ be a critical point of the energy functional, it must be a geodesic when restricted to any subinterval where it is smooth, or in other words, it must be “piecewise geodesic”. Corollary 2.7. If a piecewise smooth curve γ is a critical point of the energy functional among proper variations, then it is C 1 and thus a geodesic
7LECTURE16:VARIATIONSOFLENGTHANDENERGYProof.We can first choose proper variations with variation fields satisfyingV(t:)=0and deduce that V= 0 at any smooth point of .In particular, the first term inthe right hand of the first variation formula vanishes. As a consequence, we have(V(ti),(t) -(t)) = 0i=1for any variation field V. Then for each i we can consider all variation fields so thatV(t,)=oforall j+i,and concludethat<V(ti),(tt) -(t)) = 0口for any V(ti) e T(t). It follows that (tt) = (t) and thus is C1I Piecewise smooth curve.Finally we compute the second variation of energy.As in calculus,the secondvariation is mainly used near critical points, i.e. near geodesics. So we let : [a,b] -→M be a geodesic, and f(t, s) be a smooth variation of . According to Theorem 2.3,Proposition 3.11 and Proposition 3.13,d2ads2E(%) =(Va/atfs,ft)dtOs(Va/asVa/otfs,ft)d+(VaotfsVaasft)aaR(sfi)d+(Va/atVa/asfs,ft)d+(VaatfsVajatf)dThere are two Va/at in the above formula. We may either apply Proposition 3.11 tothefirst oneto getd2(Vpf.f)d+/R2)(f.fi)-(Vapf.apf.)+(Vaof.,Vapf.)dtds2E(b)=00of=(Vaps f.f)/a3 +)(<Ror apply Proposition 3.11 to both to getd?()+)8I()-(p.f.f)+(fp.)/a)(()a
LECTURE 16: VARIATIONS OF LENGTH AND ENERGY 7 Proof. We can first choose proper variations with variation fields satisfying V (ti) = 0 and deduce that ∇γ˙ γ˙ = 0 at any smooth point of γ. In particular, the first term in the right hand of the first variation formula vanishes. As a consequence, we have X k i=1 ⟨V (ti), γ˙(t + i ) − γ˙(t − i )⟩ = 0 for any variation field V . Then for each i we can consider all variation fields so that V (tj ) = 0 for all j ̸= i, and conclude that ⟨V (ti), γ˙(t + i ) − γ˙(t − i )⟩ = 0 for any V (ti) ∈ Tγ(ti) . It follows that ˙γ(t + i ) = ˙γ(t − i ) and thus γ is C 1 . □ ¶ Piecewise smooth curve. Finally we compute the second variation of energy. As in calculus, the second variation is mainly used near critical points, i.e. near geodesics. So we let γ : [a, b] → M be a ✿✿✿✿✿✿✿✿✿ geodesic, and f(t, s) be a smooth variation of γ. According to Theorem 2.3, Proposition 3.11 and Proposition 3.13, d 2 ds2 E(γs) = Z b a ∂ ∂s⟨∇e ∂/∂tfs, ft⟩dt = Z b a ⟨∇e ∂/∂s∇e ∂/∂tfs, ft⟩dt+ Z b a ⟨∇e ∂/∂tfs, ∇e ∂/∂sft⟩dt = Z b a ⟨Re( ∂ ∂s, ∂ ∂t)fs,ft⟩dt+ Z b a ⟨∇e ∂/∂t∇e ∂/∂sfs, ft⟩dt+ Z b a ⟨∇e ∂/∂tfs, ∇e ∂/∂tfs⟩dt. There are two ∇e ∂/∂t in the above formula. We may either apply Proposition 3.11 to the first one to get d 2 ds2 E(γs)=Z b a ∂ ∂t⟨∇e∂/∂sfs,ft⟩dt+ Z b a ⟨Re( ∂ ∂s, ∂ ∂t)fs,ft⟩−⟨∇e∂/∂sfs,∇e∂/∂tft⟩+⟨∇e∂/∂tfs,∇e∂/∂tfs⟩dt =⟨∇e∂/∂sfs,ft⟩|(b,s) (a,s)+ Z b a ⟨Re( ∂ ∂s, ∂ ∂t)fs,ft⟩−⟨∇e∂/∂sfs,∇e∂/∂tft⟩+⟨∇e∂/∂tfs,∇e∂/∂tfs⟩dt, or apply Proposition 3.11 to both to get d 2 ds2 E(γs)=Z b a ∂ ∂t ⟨∇e∂/∂sfs,ft⟩+⟨fs,∇e∂/∂tfs⟩ dt + Z b a ⟨Re( ∂ ∂s, ∂ ∂t)fs,ft⟩−⟨∇e∂/∂sfs,∇e∂/∂tft⟩−⟨fs, ∇e∂/∂t∇e∂/∂tfs⟩ dt = ⟨∇e∂/∂sfs,ft⟩+⟨fs,∇e∂/∂tfs⟩ (b,s) (a,s) + Z b a ⟨Re( ∂ ∂s, ∂ ∂t)fs,ft⟩−⟨∇e∂/∂sfs,∇e∂/∂tft⟩−⟨fs, ∇e∂/∂t∇e∂/∂tfs⟩ dt
8LECTURE16:VARIATIONSOFLENGTHANDENERGYNote that by Proposition 3.13(a),6.00()) (R(0,()(),() (, (, V)),So by letting s = 0 in both formula, we getTheorem 2.8 (The Second Variation of Energy).Let f(t,s)be a smooth variationof a geodesic : [a,b] → M, thend2E(%) = (Vansfs,)1 + / (R(6,W,V)+(V,V,VV)dtds2= (Vasfs,)+(V,V)I。 - / (V,V,V,V- R(%, V)(t)dtIn particular, if the variation is proper, then V(a) = V(b) = 0 and(Vasf.)l(a,s) = Vv(a)fs(a, s) = 0, (Valasf.)l(b,s) = Vv(b)f.(b, s) = 0so wegetd?E(%) =(R(%, V)%,V)+(V,V,V,V))dtds2(V(t), -V,V,V(t) +R(%, V)(t)) dtNotethat(R(%,V),V) = -Rm(%, V,,V) =-K(, V) Area(, V),so we see that if (M, g) has non-positive sectional curvature, then for any propervariation of any geodesics,d?2E(%) ≥ 0.ds2As a result, any geodesic in non-positive sectional curvature space is locally mini-mizingamong nearbycurves
8 LECTURE 16: VARIATIONS OF LENGTH AND ENERGY Note that by Proposition 3.13(a), ⟨Re( ∂ ∂s, ∂ ∂t)fs,ft⟩|(t,0) = ⟨R(V (t), γ˙(t))V (t), γ˙(t)⟩ = ⟨V, R( ˙γ, V ) ˙γ⟩(t). So by letting s = 0 in both formula, we get Theorem 2.8 (The Second Variation of Energy). Let f(t, s) be a smooth variation of a geodesic γ : [a, b] → M, then d 2 ds2 s=0 E(γs) = ⟨∇e∂/∂sfs,γ˙⟩|b a + Z b a ⟨R(˙γ,V) ˙γ,V ⟩+⟨∇γ˙V,∇γ˙V ⟩ dt = ⟨∇e∂/∂sfs,γ˙⟩+⟨V,∇eγ˙ V ⟩ b a − Z b a V, ∇γ˙ ∇γ˙ V − R( ˙γ, V ) ˙γ(t) dt. In particular, if the variation is proper, then V (a) = V (b) = 0 and (∇e∂/∂sfs)|(a,s) = ∇V (a)fs(a, s) = 0, (∇e∂/∂sfs)|(b,s) = ∇V (b)fs(b, s) = 0 so we get d 2 ds2 s=0 E(γs) = Z b a ⟨R( ˙γ, V ) ˙γ, V ⟩+⟨∇γ˙ V, ∇γ˙ V ⟩ dt = Z b a ⟨V (t), −∇γ˙ ∇γ˙ V (t) + R( ˙γ, V ) ˙γ(t)⟩ dt. Note that ⟨R( ˙γ, V ) ˙γ, V ⟩ = −Rm( ˙γ, V, γ, V ˙ ) = −K( ˙γ, V ) Area( ˙γ, V ), so we see that if (M, g) has non-positive sectional curvature, then for any proper variation of any geodesics, d 2 ds2 s=0 E(γs) ≥ 0. As a result, any geodesic in non-positive sectional curvature space is locally minimizing among nearby curves
LECTURE16:VARIATIONSOFLENGTHANDENERGY93.APPENDIX:THEINDUCEDCONNECTION(BYYULONGLI)I The pullback bundle.Let M,N be two smooth manifolds, V a connection on M and : N→ M asmooth map. Then we may pullback the tangent bundle : TM -→ M over M to avector bundle ' : E = β*(TM) → N over N[known as the pullback bundlel, whereE=[(r, u)rEN, UETM, (r) =π(u)) C N × TM.In other words, we just set the fiber E, of r' : E → N to be the vector space Te()M.Denote by : E -→TM the induced bundle map that maps (r,u) E to v ETM.Then wehave thefollowing commutative diagram(r, u) Umm0ETM(r, u)E9NI元NMET(u)rWW+p(a)TIn this construction, there are two natural ways to obtain sections on E:. For any section V eTo(TM), one can define an assignmentV:N-Er → (a, Ve()By this way any smooth vector field V e Fo(TM) gives rise to a smoothsection *V e F(E) of E.[Note: if E, is a local frame of TM near p(r), then *E, is a local frame of E near r.].For any section X e Fo(TN), one can define an assignmentdp(X):N→E (r, (dp)r(Xr)By this way any smooth vector field X eFo(TN) gives rise to a smoothsection dp(X) E F(E) of E.The two constructions are related as follows: For X e F(TN), V E F(TM)dp(X) = *V dpr(Xr) = Vo(r) 一 X, V are p-related.In manifold theory we have seen that if X,V and Y,W are p-related, then [X,Y]and [V,W] isp-related.SowegetProposition 3.1. If dp(X) = *V, do(Y) = *W, then dp([X,Y]) = *([V, W)
LECTURE 16: VARIATIONS OF LENGTH AND ENERGY 9 3. Appendix: The induced connection (by Yulong Li) ¶ The pullback bundle. Let M, N be two smooth manifolds, ∇ a connection on M and φ : N → M a smooth map. Then we may pullback the tangent bundle π : TM → M over M to a vector bundle π ′ : E = φ ∗ (TM) → N over N[known as the pullback bundle], where E = {(x, v)|x ∈ N, v ∈ TM, φ(x) = π(v)} ⊂ N × TM. In other words, we just set the fiber Ex of π ′ : E → N to be the vector space Tφ(x)M. Denote by φe : E → TM the induced bundle map that maps (x, v) ∈ E to v ∈ TM. Then we have the following commutative diagram (x, v) v (x, v) E TM v x N M π(v) x φ(x) ∈ ∈ ∈ π ′ φe π ∋ ∈ φ ∋ ∈ ∈ In this construction, there are two natural ways to obtain sections on E: • For any section V ∈ Γ ∞(TM), one can define an assignment φe ∗V : N → E x 7→ (x, Vφ(x)). By this way any smooth vector field V ∈ Γ ∞(TM) gives rise to a smooth section φe ∗V ∈ Γ ∞(E) of E. [Note: if Ei is a local frame of TM near φ(x), then φe ∗Ei is a local frame of E near x.] • For any section X ∈ Γ ∞(T N), one can define an assignment dφ(X) : N → E x 7→ (x, (dφ)x(Xx)). By this way any smooth vector field X ∈ Γ ∞(T N) gives rise to a smooth section dφ(X) ∈ Γ ∞(E) of E. The two constructions are related as follows: For X ∈ Γ ∞(T N), V ∈ Γ ∞(TM), dφ(X) = φe ∗V ⇐⇒ dφx(Xx) = Vφ(x) ⇐⇒ X, V are φ-related. In manifold theory we have seen that if X, V and Y, W are φ-related, then [X, Y ] and [V, W] is φ-related. So we get Proposition 3.1. If dφ(X) = φe ∗V, dφ(Y ) = φe ∗W, then dφ([X, Y ]) = φe ∗ ([V, W])
10LECTURE16:VARIATIONSOFLENGTH ANDENERGYTo extend this proposition to more general vector fields on N, we let E, be alocal frame of TM near (r),then *E, is a local frameof E near r.Proposition 3.2. If X,YeF(TN) and dp(X) =X*Er,dp(Y) =Yi*Ej, thend([X, Y])=X(Y)*E,-Y(X')*E, +X'Y([E, E,]l)Proof. For any r E N and any f e C(M),Y(*f)()=(dr)(Y)f=Y()(*E,)(f)=Y(r)(E,)()(f)=(Y*(E,f)(),so Y(β*f) = Yig*(E,f) and thusYiX(*(E,f))-X'Y(*(Ef))=XiYip*(E,E,f-E,E,f)=X'Yi*([E,E]f).It follows that as vectors in Te(r)M acting on f e C(M),(dp[X,Y)(r)f =[X,Y]r(β* f)=Xr(Y*f)-Yr(X*f)=X (Yip*(E,f))-Y (Xi*(E,f)=Xr(Yi)(E,f)+YiX((E,f))-Y(Xi)(Ef)-XiY ((E;f)=X(Y)p*(E,f)-Y(X')(Ef)+X(r)Yi(r)*([E,E]f)So we get, as sections in To(E),dp([X, Y])) = X(Y')E, -Y(X')*E, +X'Y*([Ei, E,])口I The induced connection on the pullback bundle.Since each fiber Er = Tp(r)M, one may transplant structures on TM to E. Forexample, if (M,g) is a Riemannian manifold, then the metric structure on TM givesrise to a metric structure on the pullback bundle E in the natural way, namely onejust endow each fiber E= Te(a)M with the inner product gp(a).Although it is not that obvious, we may also transplant linear connections onTM to E:Proposition 3.3. Given any linear connection on TM, there erists a uniquelinearconnectionVonE satisfyingu(*V) =*(V(dp)uV)(1)for any rE N, uETN and VEFo(TM).Proof. We first prove the uniqueness. Assume exists. For any ro e N and anylocal frame (E,jm1 around (ro), (*E,}m, is a local frame around ro. Thus forany s E(E), we may writes(r) = s(r)p*E;(r)
10 LECTURE 16: VARIATIONS OF LENGTH AND ENERGY To extend this proposition to more general vector fields on N, we let Ei be a local frame of TM near φ(x), then φe ∗Ei is a local frame of E near x. Proposition 3.2. If X, Y ∈Γ ∞(T N) and dφ(X) = Xiφe ∗Ei , dφ(Y ) = Y jφe ∗Ej , then dφ([X, Y ]) = X(Y j )φe ∗Ej − Y (X i )φe ∗Ei + X iY jφe ∗ ([Ei , Ej ]). Proof. For any x ∈ N and any f ∈ C ∞(M), Y (φ ∗ f)(x)= (dφx)(Yx)f =Y j (x)(φe ∗Ej )x(f)=Y j (x)(Ej )φ(x)(f)= (Y jφ ∗ (Ejf))(x), so Y (φ ∗ f) = Y jφ ∗ (Ejf) and thus Y jXx φ ∗ (Ejf) −X iYx φ ∗ (Eif) =X iY jφ ∗ (EiEjf −EjEif)=X iY jφ ∗ ([Ei , Ej ]f). It follows that as vectors in Tφ(x)M acting on f ∈ C ∞(M), (dφ[X,Y ])φ(x)f =[X,Y ]x(φ ∗ f) =Xx(Y φ∗ f)−Yx(Xφ∗ f) =Xx Y jφ ∗ (Ejf) −Yx X iφ ∗ (Eif) =Xx(Y j )φ ∗ (Ejf)+Y jXx φ ∗ (Ejf) −Yx(X i )φ ∗ (Eif)−X iYx φ ∗ (Eif) =Xx(Y j )φ ∗ (Ejf)−Yx(X i )φ ∗ (Eif)+X i (x)Y j (x)φ ∗ ([Ei , Ej ]f). So we get, as sections in Γ∞(E), dφ([X, Y ]) = X(Y j )φe ∗Ej − Y (X i )φe ∗Ei + X iY jφe ∗ ([Ei , Ej ]). □ ¶ The induced connection on the pullback bundle. Since each fiber Ex = Tφ(x)M, one may transplant structures on TM to E. For example, if (M, g) is a Riemannian manifold, then the metric structure on TM gives rise to a metric structure on the pullback bundle E in the natural way, namely one just endow each fiber Ex = Tφ(x)M with the inner product gφ(x) . Although it is not that obvious, we may also transplant linear connections on TM to E: Proposition 3.3. Given any linear connection ∇ on TM, there exists a unique linear connection ∇e on E satisfying (1) ∇e u(φe ∗V ) = φe ∗ (∇(dφ)xuV ) for any x ∈ N, u ∈ TxN and V ∈ Γ ∞(TM). Proof. We first prove the uniqueness. Assume ∇e exists. For any x0 ∈ N and any local frame {Ei} m i=1 around φ(x0), {φe ∗Ei} m i=1 is a local frame around x0. Thus for any s ∈ Γ ∞(E), we may write s(x) = s i (x)φe ∗Ei(x)