《黎曼几何》课程教学资源（英文讲义）Lec16 VARIATIONS OF LENGTH AND ENERGY.pdf

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

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

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

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