LECTURE 17: JACOBI FIELDS As we have seen, in the second variational formula the curvature term appears. As a result, the formula will play a crucial role in studying the relation between curvature and topology of Riemannian manifolds. Usually the first step will be: start with a geodesic, and take a special variation (e.g. a geodesic variation, sometimes with one endpoint fixed). Thus the variation field of a geodesic variation will be very important for the remaining of this course. 1. Definition of the Jacobi field ¶ The Jacobi field. Let γ be a geodesic in (M, g). Suppose f : [a, b] × (−ε, ε) → M is a ✿✿✿✿✿✿✿✿ geodesic ✿✿✿✿✿✿✿✿✿ variation of γ, i.e. each curve γs = f(·, s) is a geodesic in M. Then for any s, ∇e ∂/∂tft = ∇e ∂/∂tγ˙ s = 0. As a consequence, ∇e ∂/∂t∇e ∂/∂tfs = ∇e ∂/∂t∇e ∂/∂sft = ∇e ∂/∂t∇e ∂/∂sft − ∇e ∂/∂s∇e ∂/∂tft = Re( ∂ ∂t, ∂ ∂s)ft . Taking s = 0, we see that the variation field V of any geodesic variation satisfies (1) ∇γ˙ ∇γ˙ V = R( ˙γ, V ) ˙γ. Definition 1.1. Let X be a smooth vector field X along a geodesic γ. We call X a Jacobi field along γ if the equation (1) holds. Remark. Let γ be a geodesic. There are two trivial Jacobi fields along γ: • Obviously X = ˙γ is a Jacobi field. It is the variation field of f(t, s) = γ(t+s). • X = tγ˙ is a Jacobi field since ∇γ˙ ∇γ˙(tγ˙) = ∇γ˙( ˙γ + t∇γ˙ γ˙) = 0 and R( ˙γ, tγ˙) ˙γ = 0. It is the variation field of f(t, s) = γ(t + st). • But X = t 2γ˙ is NOT a Jacobi field since ∇γ˙ ∇γ˙(t 2 γ˙) = ∇γ˙(2tγ˙) = 2 ˙γ ̸= 0. It is not amazing that t 2γ˙ is no longer a Jacobi field along γ: Lemma 1.2. Let X be a Jacobi field along γ, then f(t) = ⟨X, γ˙⟩ is a linear function. 1
2 LECTURE 17: JACOBI FIELDS Proof. According to the Jacobi field equation, f ′′(t) = d 2 dt2 ⟨X, γ˙⟩ = ⟨∇γ˙ ∇γ˙ X, γ˙⟩ = ⟨R( ˙γ, X) ˙γ, γ˙⟩ = 0. It follows that ⟨X, γ˙⟩ is a linear function along γ. □ ¶ Existence and uniqueness of Jacobi field. So the variation field of any geodesic variation is a Jacobi fields. As a result, the second variation formula for a geodesic variation is very simple. We will show that conversely, any Jacobi field on γ can be realized as the variation field of some geodesic variation of γ. Before we prove it, we need some basic properties of Jacobi fields. Let’s take a closer look of the equation for Jacobi fields. Since it is a differential equation, it is enough to study the equation in a coordinate chart. Although one may work on a general frame, to simply the computation one may use a special frame that are parallel along γ [so that the covariant derivatives of the frame are as simple as possible]. So we start with an orthonormal basis {e1, · · · , em} of TpM, with e1 = ˙γ(a), where p = γ(a). Let ei(t) := the parallel transport of ei along γ, 1 ≤ i ≤ m. According to Proposition 2.1 in Lecture 6, ⟨ei(t), ej (t)⟩γ(t) = ⟨ei , ej ⟩γ(a) = δij . In other words, we get an orthonormal frame {e1(t), · · · , em(t)} along γ with e1(t) = γ˙(t), and this frame is parallel along γ, i.e. ∇γ˙ (t)ek(t) = 0, 1 ≤ k ≤ m. Let X be a Jacobi field along γ, then with respect to this orthonormal frame we can write X = Xi (t)ei(t), and we get ∇γ˙ X = X˙ i (t)ei(t) and ∇γ˙ ∇γ˙ X = X¨i (t)ei(t). It follows that the Jacobi field equation becomes X¨i (t)ei(t) − X i (t)R j 1i1 ej (t) = 0. So we arrived at a system of second order homogeneous ODEs, X¨i (t) − X j (t)R i 1j1 = 0, 1 ≤ i ≤ m, Using the basic theory for second order homogeneous ODEs, we get Theorem 1.3. Let γ : [a, b] → M be a geodesic, then for any Xγ(a) , Yγ(a) ∈ Tγ(a)M, there exists a unique Jacobi field X along γ so that X(a) = Xγ(a) and ∇γ˙ (a)X = Yγ(a) . Moreover, the set of Jacobi fields along γ is a linear space of dimension 2m (which is canonically isomorphic to Tγ(a)M ⊕ Tγ(a)M)
LECTURE 17: JACOBI FIELDS 3 The following consequence is fundamental: Corollary 1.4. If X(t) is a Jacobi field along γ, and X is not identically zero, then the zeroes of X are discrete. Proof. If X has a sequence of zeroes that converges to t0, then X1 (t) = · · · = x m(t) = 0 for a sequence of points tk converging to γ(t0). It follows that Xi (t0) = 0 and X˙ i (t0) = 0 for all i, i.e. X(t0) = 0, ∇γ˙ (t0)X = 0. By uniqueness, X ≡ 0. □ ¶ Jacobi fields as variational fields of geodesic variation. Now we prove that each Jacobi field X along a geodesic γ can be realized as the variation field of a geodesic variation of γ(So the space of all the Jacobi fields along γ describes all possible ways that γ can vary in “the space of all geodesics” infinitesimally): Theorem 1.5. A vector field X along a geodesic γ is a Jacobi field if and only if X is the variation field of some geodesic variation of γ. Proof. We have seen that the variation field of any geodesic variation of γ is a Jacobi field. Now we suppose X is a Jacobi field along γ and construct the desired geodesic variation. For simplicity we parameterize γ as γ : [0, 1] → M, so γ(t) = expγ(0)(tγ˙(0)) is defined for 0 ≤ t ≤ 1. It follows that for any (p, Yp) in a small neighborhood of (γ(0), γ˙(0)), the exponential map expp (tYp) is defined for 0 ≤ t ≤ 1. Let ξ : (−ε, ε) → M be the geodesic with initial conditions ξ(0) = γ(0), ˙ξ(0) = Xγ(0). Let T(s), W(s) be parallel vector fields along ξ with T(0) = ˙γ(0) and W(0) = ∇γ˙ (0)X. Define f : [0, 1] × (−ε, ε) → M by f(t, s) = expξ(s) (t(T(s) + sW(s))). As we mentioned above, for ε small enough, f is well-defined. Moreover, f(t, 0) = γ(t), so f is a geodesic variation of γ. Let V be the variation field of f. Since both V and X are Jacobi fields along γ, to show V = X, it is enough to show V (0) = Xγ(0) and ∇γ˙ (0)V = ∇γ˙ (0)X. The first one follows from V (0) = fs(0, 0) = d ds s=0 f(0, s) = d ds s=0 ξ(s) = Xγ(0). For the second one, we start with the fact ∇e ∂/∂tfs = ∇e ∂/∂sft . Evaluate the left hand side at (0, 0) we get ∇e ∂/∂tfs 0,0 = ∇e ∂/∂tfs(t, 0) t=0 = ∇γ˙ (0)V
4 LECTURE 17: JACOBI FIELDS and evaluate the right hand side at (0, 0) and use the fact ft(0, s) = (d expξ(s) )0 d dt t=0 (t(T(s) + sW(s))) = T(s) + sW(s) we get ∇e ∂/∂sft 0,0 = ∇e ∂/∂sft(0, s) s=0 = ∇Xγ(0) (T(s) + sW(s)) = W(0) = ∇γ˙ (0)X. So we get ∇γ˙ (0)V = ∇γ˙ (0)X and thus completes the proof. □ Note that given any Jacobi field V along a geodesic γ, there exist many geodesic variations of γ whose variation fields are V [analogue: given any vector v at a point p, there exist many curves whose tangent vector at p is v]. In the proof above we give an explicit formula for one such geodesic variations, namely, (2) f(t, s) = expξ(s) (t(T(s) + sW(s))), where ξ is a geodesic with ξ(0) = γ(0) and ˙ξ(0) = V (0), and T, W are parallel vector fields along ξ with T(0) = ˙γ(0) and W(0) = ∇γ˙ (0)V . 2. Jacobi fields with special conditions ¶ Normal Jacobi fields. The obviously Jacobi fields ˙γ, tγ˙ [and their linear combinations] along γ are both ✿✿✿✿✿✿✿✿ tangent to γ and are not so interesting in applications. Very often we need to rule out them and mainly consider ✿✿✿✿✿✿✿ normal Jacobi fields. Definition 2.1. A Jacobi field along γ is called a normal Jacobi field if it is perpendicular to ˙γ along γ. It turns out that for any Jacobi field, the tangential components must be a linear combination of ˙γ and tγ˙ : Proposition 2.2. For any Jacobi field X along γ, there exists c 1 , d1 ∈ R so that X ⊥ = X − c 1 tγ˙ − d 1 γ˙ is a normal Jacobi field along γ. Proof. By Lemma 1.2, ⟨X, γ˙⟩ is a linear function along γ, i.e. ⟨X, γ˙⟩ = c1t + d1 for some constant c1, d1 ∈ R. Now we let X ⊥ = X − c 1 tγ˙ − d 1 γ˙ with c 1 = c1 |γ˙ | 2 , d1 = d1 |γ˙ | 2 . Then it is a Jacobi field along γ since it is a linear combination of Jacobi fields along γ, and it is normal since ⟨X ⊥, γ˙⟩ = c1t + d1 − c 1 t|γ˙ 2 | − d 2 |γ˙ | 2 = 0
LECTURE 17: JACOBI FIELDS 5 □ Note that if X⊥ is a normal Jacobi field along γ, then ⟨∇γ˙ X ⊥, γ˙⟩ = d dt⟨X ⊥, γ˙⟩ − ⟨X ⊥, ∇γ˙ γ˙⟩ = 0 and thus ∇γ˙ X⊥ ⊥ γ˙ . It follows Corollary 2.3. A Jacobi field X along γ is normal if and only if ⟨X(a), γ˙(a)⟩ = ⟨∇γ˙ (a)X, γ˙(a)⟩ = 0. In particular, the set of normal Jacobi fields form a linear space of dimension 2m−2. Proof. With X = X⊥ + c 1 tγ˙ + d 1γ˙ , we have ⟨X(a), γ˙(a)⟩ = (c 1 a + d 1 )|γ˙ | 2 , ⟨∇γ˙ (a)X, γ˙(a)⟩ = ⟨∇γ˙ (a)(c 1 tγ˙ + d 1 γ˙), γ˙(a)⟩ = c 1 |γ˙ | 2 . The conclusion follows. □ Corollary 2.4. Let X be a Jacobi field so that ⟨X(t1), γ˙(t1)⟩ = ⟨X(t2), γ˙(t2)⟩ = 0 for two distinct numbers t1, t2. Then X is a normal Jacobi field. Proof. This follows from Lemma 1.2, i.e. ⟨X, γ˙⟩ is a linear function along γ, and the fact that a linear function has no more than one zero unless it is identically zero. □ ¶ Normal Jacobi fields on spaces with constant sectional curvature. Let (M, g) be a Riemannian manifold with constant sectional curvature k, i.e. R(X, Y )Z = −k(⟨X, Z⟩Y − ⟨Y, Z⟩X). Let γ be a normal geodesic in M, and X a normal Jacobi field along γ. Then R( ˙γ, X) ˙γ = −k(⟨γ, ˙ γ˙⟩X − ⟨X, γ˙⟩γ˙) = −kX. So the equation for a normal Jacobi field X along γ becomes ∇γ˙ ∇γ˙ X + kX = 0. Again we take an orthonormal frame {ei(t)} along γ so that • e1(t) = ˙γ(t), • each ei(t) is parallel along γ, as we did in the proof of Theorem 1.3, and write X = Xm i=2 X i (t)ei(t)
6 LECTURE 17: JACOBI FIELDS then the equation for the coefficient Xi (t) becomes X¨i (t) + kXi (t) = 0, 2 ≤ i ≤ m. The solution to this equation is X i (t) = c i sin(√ kt) √ k + d i cos(√ kt), if k > 0, c i t + d i , if k = 0, c i sinh(√ −kt) √ −k + d i cosh(√ −kt), if k 0 t, k = 0 sinh(√ −kt) √ −k , k 0 1, k = 0 cosh(√ −kt), k < 0 so that we can write Xi (t) = c i snk(t) + d i cnk(t). ¶ Jacobi fields with V (0) = 0. For simplicity let a = 0 for the defining interval [a, b] of γ. In many applications we need geodesic variations that fix one end, i.e. with γs(0) = γ(0) for all s. Of course the Jacobi field for such geodesic variations satisfies V (0) = 0. Conversely, if V is a Jacobi field along γ with V (0) = 0, then in (2) we may take ξ(s) ≡ γ(0), T(s) ≡ γ˙(0), W(s) ≡ ∇γ˙ (0)V and get an explicit geodesic variation with one end fixed, whose variation field is V : Proposition 2.5. If V is a Jacobi field along geodesic γ with V (0) = 0, then f(t, s) = expγ(0)(t( ˙γ(0) + s∇γ˙ (0)V )). is a geodesic variation of γ with γs(0) = γ(0) and whose variation field is V . In particular, by calculating the variation field of the above variation via its formula, we get Corollary 2.6. If V is a Jacobi field along geodesic γ with V (0) = 0, then V (t) = fs(t, 0) = (d expγ(0))tγ˙ (0)(t∇γ˙ V )
LECTURE 17: JACOBI FIELDS 7 ¶ Taylor’s expansion of the Jacobi field with V (0) = 0. Now let V, W be Jacobi fields along a geodesic γ with V (0) = 0, ∇γ˙ (0)V = Xp ∈ TpM and W(0) = 0, ∇γ˙ (0)V = Yp ∈ TpM. According to Corollary 2.6, we have V (t) = (d expp )tγ˙ (0)(tXp) and W(t) = (d expp )tγ˙ (0)(tYp). Let f(t) = ⟨V (t), W(t)⟩. Then we have f(0) = ⟨V (0), W(0)⟩ = 0, f ′ (0) = ⟨∇γ˙ (0)V, W(0)⟩ + ⟨V (0), ∇γ˙ (0)W⟩ = 0, f ′′(0) = ⟨∇γ˙ (0)∇γ˙ V, W(0)⟩ + 2⟨∇γ˙ (0)V, ∇γ˙ (0)W⟩ + ⟨V (0), ∇γ˙ (0)∇γ˙W⟩ = 2⟨Xp, Yp⟩. To compute more derivatives, we note that in view of V (0) = 0, ∇γ˙ (0)∇γ˙ V = R( ˙γ(0), V (0)) ˙γ(0) = 0, and similarly ∇γ˙ (0)∇γ˙W = 0. So [We abbreviate the k th composition ∇γ˙ · · · ∇γ˙ to ∇ (k) γ˙ ] f ′′′(0) = X 3 l=0 3 l ⟨∇(3−l) γ˙ V, ∇ (l) γ˙ W⟩(0) = 0, f ′′′′(0) = X 4 l=0 4 l ⟨∇(4−l) γ˙ V, ∇γ˙W⟩(0) = 4⟨∇(3) γ˙ V, ∇γ˙W⟩(0) + 4⟨∇γ˙ V, ∇ (3) γ˙ W⟩(0). To calculate the third order derivative, we note that if we take the (k−2)th covariant derivative of the Jacobi field equation for V , then ∇ (k) γ˙ V − X k−2 l=0 k − 2 l (∇ (k−2−l) γ˙ R)( ˙γ, ∇ (l) γ˙ V ) ˙γ = 0, where we used the facts ∇W (R(X, Y )Z)= (∇W R)(X, Y )Z+R(∇W X, Y )Z+R(X, ∇W Y )Z+R(X, Y )∇W Z and ∇γ˙ γ˙ = 0. Taking k = 3, we get ∇ (3) γ˙ V − (∇γ˙R)( ˙γ, V ) ˙γ − R( ˙γ, ∇γ˙ V ) ˙γ = 0. Evaluate at t = 0, and use V (0) = 0, we get (∇ (3) γ˙ V )(0) = R( ˙γ(0), Xp) ˙γ(0). Thus f ′′′′(0)=4⟨R( ˙γ(0), Xp) ˙γ(0), Yp⟩+4⟨Xp, R( ˙γ(0), Yp) ˙γ(0)⟩=−8Rm( ˙γ(0), Xp, γ˙(0), Yp). So we get ⟨V (t), W(t)⟩ = ⟨Xp, Yp⟩t 2 − 1 3 Rm( ˙γ(0), Xp, γ˙(0), Yp)t 4 + O(t 5 ). In particular, if we take W = V and assume |Xp| = 1, then |V (t)| 2 = t 2 − 1 3 Rm( ˙γ(0), Xp, γ˙(0), Xp)t 4 + O(t 5 )