LECTURE 12: VARIATIONS AND JACOBI FIELDS 1. Formulas for the first and second variations Definition 1.1. Let γ :[a; b] ! M be a smooth curve, and " > 0. (1)A variation of γ is a smooth map f :[a; b] × (−"; ") ! M so that f(t; 0) = γ(t) for all t 2 [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 2 (−"; "), γs(a) = γ(a) and γs(b) = γ(b): (3) The variation is called a geodesic variation if each γs is a geodesic. Let f be a variation of γ. For simplicity we will denote @ @ f := df( ); f := df( ): s @s t @t Note that they are both vector fields near γ. Lemma 1.2. rfs ft = rft fs: Proof. @ @ r f − r f = [f ; f ] = df([ ; ]) = 0: fs t ft s s t @s @t By definition, ft =γ _ s: In particular, on γ one has ft(γ(t)) =γ: _ Definition 1.3. We will call V (t) = fs(γ(t)) the variation field of f along γ. [It is a vector field along γ.] Last time we calculated the first and second variation of the energy functional 1 Z b E(γ) = jγ_ (t)j2dt 2 a with respect to proper variations in local charts, assuming that the whole family of curves sit in one chart. Now we will give a re-formulation of that formula without these assumptions and give an invariant proof. 1 2 LECTURE 12: VARIATIONS AND JACOBI FIELDS Theorem 1.4 (The First Variation of Energy). Let f(t; s) be a smooth variation of a smooth curve γ. Then Z b dE(γs) = hfs; fti(b; s) − hfs; fti(a; s) − hfs; rft ftidt: ds a As a consequence, Z b dE(γs) (0) = − hV (t); rγ_ γ_ i dt − hV (a); γ_ (a)i + hV (b); γ_ (b)i : ds a In particular, if f is also a proper variation, then Z b dE(γs) (0) = − hV (t); rγ_ γ_ i dt: ds a Proof. The derivative of E(γs) is Z b dE(γs) 1 @ = hγ_ s(t); γ_ s(t)idt ds 2 a @s 1 Z b = rfs hft; ftidt 2 a Z b = hrfs ft; ftidt a Z b = hrft fs; ftidt a Z b @ Z b = hfs; ftidt − hfs; rft ftidt a @t a Z b = hfs; fti(b; s) − hfs; fti(a; s) − hfs; rft ftidt: a Use the same way, one can calculate the first variation of the length. A trick to simplify the computation is the following observation: @ @ 1 1 1 @ 1 ft 2 jγ_ s(t)j = hft; fti = hft; fti = hrft fs; fti = hrγ_ (t)fs; i: @s @s 2 jftj @s jftj jftj Then following the same computation, one gets Theorem 1.5 (The First Variation of Length). Let f(t; s) be a smooth variation of a smooth curve γ. Then Z b dL(γs) γ_ γ_ (a) γ_ (b) (0) = − V (t); rγ_ dt − V (a); + V (b); : ds a jγ_ j jγ_ (a)j jγ_ (b)j LECTURE 12: VARIATIONS AND JACOBI FIELDS 3 More generally, one can consider piecewise smooth curves γ :[a; b] ! M. More precisely, 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 are continuous functions f :[a; b] × (−"; ") ! M so that f is smooth on each [ti; ti+1] × (−"; ") for each i, and so that fs is well defined even at ti's. Applying the previous theorems to each [ti; ti+1] × (−"; "), we get Corollary 1.6. Let f be a variation of a piecewise smooth curve γ. Then Z b k dE(γs) X (0) = − hV (t); r γ_ i dt−hV (a); γ_ (a)i+hV (b); γ_ (b)i− hV (t ); γ_ (t+)−γ_ (t−)i ds γ_ i i i a i=1 and Z b dL(γs) γ_ γ_ (a) γ_ (b) (0) = − V (t); rγ_ dt − V (a); + V (b); ds a jγ_ j jγ_ (a)j jγ_ (b)j k X γ_ (t+) γ_ (t−) − V (t ); i − i : i jγ_ (t+)j jγ_ (t−)j i=1 i i Last time we only showed that among smooth curves, geodesics are critical points of the energy functional. A natural question is: If a curve is not smooth, but 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 1.7. If a piecewise smooth curve γ is a critical point of the energy func- tional, then it is C1 and thus a geodesic. Proof. We can first choose proper variations with variation fields satisfying V (ti) = 0 and deduce that rγ_ 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 k X + − hV (ti); γ_ (ti ) − γ_ (ti )i = 0 i=1 for any variation field V . Then for each i we can consider all variation fields so that V (tj) = 0 for all j 6= i, and conclude that + − hV (ti); γ_ (ti ) − γ_ (ti )i = 0 for any V (ti) 2 Tγ(ti). It follows that + − γ_ (ti ) =γ _ (ti ); 1 and thus γ is C . 4 LECTURE 12: VARIATIONS AND JACOBI FIELDS Next we will give an invariant proof for the second variation of energy without restricting ourself to one coordinate chart. As in calculus, the second variation is mainly used near critical points, i.e. near geodesics. Theorem 1.8 (The Second Variation of Energy). Let γ :[a; b] ! M be a geodesic, and f(t; s) be a smooth variation of γ. Then 2 Z b d E(γs) 2 (0) = − hV (t); rγ_ rγ_ V (t) + R(_γ; V )_γ(t)i dt ds a − hV (a); rγ_ V (a)i + hV (b); rγ_ V (b)i − hrV (a)fs; γ_ (a)i + hrV (b)fs; γ_ (b)i: In particular, if the variation is proper, then V (a) = V (b) = 0, and we have 2 Z b d E(γs) 2 (0) = − hV (t); rγ_ rγ_ V (t) + R(_γ; V )_γ(t)i dt: ds a Proof. According to the previous computation, 2 Z b d E(γs) @ 2 = hrft fs; ftidt ds a @s Z b Z b = hrfs rft fs; ftidt + hrft fs; rfs ftidt a a Z b Z b Z b = − hR(fs; ft)fs; ftidt + hrft rfs fs; ftidt + hrft fs; rft fsidt a a a Z b = − hR(ft; fs)ft + rft rft fs; fsidt a Z b @ Z b + (hrfs fs; fti + hrft fs; fsi) dt − hrfs fs; rft ftidt: a @t a Letting s = 0, we get the formula we want. Remark. Similarly one can write down a formula for the second variation of length, or a formula for \piecewise smooth" variation. If we take s = 0 in the third line of the computation above, we will get the following alternative formula: 2 Z b d E(γs) 2 (0) = − (hV (t);R(_γ; V )_γ(t)i − hrγ_ V; rγ_ V i) dt ds a − hrV (a)fs; γ_ (a)i + hrV (b)fs; γ_ (b)i: As a consequence, we have Corollary 1.9. If (M; g) is a Riemannian manifold with non-positive sectional cur- vature, then any geodesic is locally minimizing. LECTURE 12: VARIATIONS AND JACOBI FIELDS 5 2. The Jacobi field Now suppose γ is a geodesic and f is a geodesic variation of γ, i.e. each γs = f(·; s) is a geodesic. Let V be its variation field. Then rft ft = rγ_s γ_ s = 0 and @ @ [f ; f ] = df[ ; ] = 0; t s @t @s so we get rft rft fs = rft rfs ft = −∇fs rft ft + rft rfs ft + r[fs;ft]ft = R(fs; ft)ft: Taking s = 0, we see rγ_ rγ_ V + R(_γ; V )_γ = 0: Definition 2.1. A vector field X along a geodesic γ is called a Jacobi field if rγ_ rγ_ X + R(_γ; X)_γ = 0: So the variation field of a geodesic variation is a Jacobi field. We will see later that any Jacobi field can be realized as the variation field of some geodesic variation. Example. Let γ be a geodesic. (1) Obviously X =γ _ is a Jacobi field. (2) X = tγ_ is a Jacobi field since rγ_ rγ_ (tγ_ ) = rγ_ [_γ + trγ_ γ_ ] = 0 and R(_γ; tγ_ )_γ = 0: (3) But X = t2γ_ is NOT a Jacobi field since 2 rγ_ rγ_ (t γ_ ) = rγ_ (2tγ_ ) = 2γ _ 6= 0: Theorem 2.2. Let γ :[a; b] ! M be a geodesic, then for any Xγ(a);Yγ(a) 2 Tγ(a)M, there exists a unique Jacobi field X along γ so that X(a) = Xγ(a) and rγ_ (a)X = Yγ(a): Proof. Without loss of generality, we may assume that γ is parametrized by arc length. Let fei(t)g be orthonormal basis at each point γ(t) with each ei(t) parallel along γ, and so that e1(t) =γ _ (t). Note that rγ_ (t)ek(t) = 0 i for all k. So for a vector field X = X (t)ei(t) along γ, _ i ¨ i rγ_ X = X (t)ei(t) and rγ_ rγ_ X = X (t)ei(t): 6 LECTURE 12: VARIATIONS AND JACOBI FIELDS It follows that the Jacobi field equation becomes ¨ i i j X (t)ei(t) + X (t)R1i1 ej(t) = 0: This is equivalent to ¨ i j i X (t) + X (t)R1j1 = 0; 1 ≤ i ≤ m; which is a system of second order linear ODEs.