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1 Preliminary Technical Remark 2 Geodesics, Exponential Map, Hopt Math 6397 Riemannian Geometry II The Jacobi Field 1 Preliminary Technical Remark We start with a preliminary technical remark. Let (M, g) be a m-dimensional manifold with Levi-Civita connection 5. Let H be either I or I × I, where I is an interval. Let f : H → M be a smooth map. We consider the bundle f ∗(TM) over H, and we introduce a connection f ∗5 on f ∗(TM) by putting, ∗ for every X ∈ TxH, Y a section f (TM), ∗ (f 5)X Y := 5f∗(X)Y. In the sequel, instead of (f ∗5), we shall simply write 5, since the map will be clear from the context. A section of f ∗(TM) is called a vector fields along f. An important role will be played by vector fields along γ : I → M, i.e. section of γ∗(TM). 2 Geodesics, Exponential map, Hopt-Rinow theorem Let (M, g) be an m-dimensional Riemannian manifold and 5 the Riemannian connection. A differentiable curve γ :[a, b] → M on M is geodesic in M 0 if the tangent vector field γ (t) = γ∗(d/dt) is parallel along the curve γ, i.e. 0 i 5γ0(t)γ (t) = 0. Suppose that γ is given in a coordinate chart (U, x ) by xi = xi ◦ γ(t), 1 ≤ i ≤ m. Then γ is geodesic if and only if 2 i m j k d x X i dx dx 2 + Γkj = 0 1 ≤ i ≤ m. dt j,k=1 dt dt Since 0 0 2 0 0 0 2 0 0 (d/dt) < γ (t), γ (t) > := γ (t) < γ (t), γ (t) > = 2 < 5γ0(t)γ (t), γ (t) >= 0, the length kγ0(t)k is independent of t. Thus Z t s(t) = kγ0(u)kdu = kγ0(t)k(t − a), a i.e., the arc-length s(t) of γ is a linear function of the parameter t. The parameter t is actually arc-length if and only if kγ0(t)k = 1, in which we say that the geodesic is normalized. 1 According the theorem about existence of the solution for (system) ODEs, given a point P ∈ M and v ∈ TP (M) with kvk < δ (some positive con- stant), there exists a unique geodesics C(t, P, v):(−, ) → M (or we just 0 write Cv(t)) such that Cv(0) = P , Cv(0) = v. We make the following very important remark: we always have Cλv(t) = Cv(λt), because if we set α(u) = Cv(λu), then dα dC d2α d2C = λ v , = λ2 v . du dt du2 dt2 Thus, since Cv is a geodesic, α is also a geodesic and satisfies that α(0) = 0 P, α (0) = λv, so by the uniqueness, Cv(λt) = Cλv(t). This proves the re- mark. Note that Cv(λt) is defined on (−/λ, /λ). The remark permits us to make the interval of definition of a geodesic uniformly large in a neigh- borhood of P ∈ M, say on (−2, 2). So we can define the exponential map expp : TpM → M (note that expp may not be defined on the whole space of TpM) by v 7→ γ(1; p, v), v ∈ TpM, where γv(t) := γ(t; q, v) (or we just write γ(t)) defined on (−2, 2) is the 0 unique geodesic that satisfies γ(0) = p, γ (0) = v. Geometrically, expp(v) is a point of M obtained by going out the length equal to kvk, starting from p, along the geodesic γ which passes through p with velocity equal to v/kvk(note that γv/kvk(kvk) = γv(1)). Let us calculate (exp)∗0. Let I0 : TpM → T0(TpM) be the canonical d isomorphisim, i.e. I0(v) = dt (tv)|t=0. For, v ∈ B(0, ) ⊂ TpM, recall that γv(t) = γtv(1), then d (exp) (v) := (exp) (I (v)) = exp (tv)| ∗0 ∗0 0 dt p t=0 d = γ (1)| dt tv t=0 d = γ (t)| dt v t=0 = v. So (exp)∗0 is the identity map from Tp(M) to Tp(M). In particular, it is non-singular at 0. By the inverse function theorem, it implies that expp is a 2 local diffeomorphism in a neighborhood of 0 on TpM, i.e., there exists > 0 such that expP : B(0p, ) ⊂ TP (M) → M is a diffeomorphism onto its image. Recall that TM is the set of pairs (p, v) where p ∈ M and v ∈ TpM. If {xi} are the local coordinates around p, then v = yi∂/∂xi. Thus, we can take (x1, . , xm, y1, . , ym) as coordinates of (p, v) in TM. In what follows we identify the tangent space T (TMp)v to TMp at v ∈ TpM with TpM itself by identifying ∂/∂yi with ∂/∂xi. Lemma (Gauss). Let p ∈ M and v ∈ TpM such that exppv is defined. Let ∼ w ∈ TpM = T (TpM)v. Then < (expp)∗v(v), (expp)∗v(w) >=< v, w > . The proof uses (at least implicitly) the theory of Jacobi field, so it will be proved in the next section. Hopf-Rinow Theorem. Let (M, g) be a complete Riemannian manifold. Then for every p ∈ M, the exponential map expp is defined everywhere in TpM. Proof. If not, i.e. there is some point p ∈ M and unit-vector v ∈ TpM such that the geodesic γ(t) = expp(tv) is not defined at some t0 > 0. WLOG, we assume that expp(tv) is defined on [0, t0). Take a increasing ti ∈ [0, t0) such that ti → t0. Then, for every > 0, there exists N such that for i ≥ j > N, 0 ≤ ti − tj < . Since the geodesic γ has arc-length as its parameter, d(γ(ti), γ(tj)) ≤ L(γ|[ti,tj ]) = ti − tj < . So {γ(tj)} is a Cauchy sequence. By the complete assumption, it conver- gences to q ∈ M. So limt→t0 γ(t) = q. Define γ(t0) = q, then the definition of γ can be extended across t0, which contradicts the choice of t0. Corollary. At every two points p, q in a complete Riemannian manifold, there exists a geodesic which connects p and q. 3 3 Jacobi field Let γ :[a, b] → M be a geodesic, and J = J(t) be a tangent vector field along γ. Denote by 0 5J J (t) = 5 ∂ J = 5γ0 J = , ∂t dt 2 00 5 J J (t) = 5 ∂ 5 ∂ J = 5γ0 5γ0 J = . ∂t ∂t dt2 Definition. Let γ :[a, b] → M be a geodesic on an m-dimensional Riemma- nian manifold (M, g). A vector field J = J(t) along γ is called Jacobi field if it satisfies the following Jacobi equation: J 00(t) + R(J(t), γ0(t))γ0(t) = 0. It is obvious that if J is a Jacobi field along γ, then J ⊥ := J− < J, T > T is also a Jacobi field along γ where T = γ0. 0 Using the compatibility with the metric and the fact that 5 ∂ γ (t) = 0, ∂t we compute d2 < J, γ0 > = < J 00(t), γ0 > dt2 = − < R(J, γ0)γ0, γ0 > = −R(J, γ0, γ0, γ0) = 0, by the symmetries of the curvature tensor. Thus, the Jacobi field J has the following important property: d2 < J, γ0 >≡ 0 dt2 so < J(t), γ0(t) > is a linear function of t, i.e. < J(t), γ0(t) >= λt + µ for some constants λ and µ. Next, we want to find a local expression the the Jacobi equation. Let γ be a geodesic on M. We express the Jacobi field equation in terms of the 4 local coordinate. Choose an orthonormal basis {ei|p} for TpM, and extend it to a parallel orthonormal frame along all γ. Then 0 ei(t) = 5 ∂ ei(t) = 0. ∂t Let J(t) be a Jacobi field along γ. Write m X i J(t) = J (t)ei(t). i=1 Then m m 0 X i0 00 X i00 J (t) = J (t)ei(t),J (t) = J (t)ei(t). i=1 i=1 0 Pm 0k Write γ (t) = k=1 γ ek, then Thus 00 0 0 X i00 X i j 0k 0l J (t) + R(J(t), γ (t))γ (t) = {J (t) + Rljk(γ(t))J γ γ }ei(t). i j Hence the Jacobi equation is reduced to i00 X i j 0k 0l J (t) = Rljk(γ(t))J γ γ , 1 ≤ im. j Theorem 3.1. Let γ :[a, b] → M be a geodesic on an m-dimensional Riemmanian manifold (M, g). Then for every v, w ∈ Tγ(a)M, there exists a unique Jacobi field J(t) along γ satisfying J(a) = v, J 0(a) = w. There are always two trivial Jacobi fields along any geodesic γ, which can 0 0 0 0 be written down immediately: Because 5 ∂ γ (t) = 0 and R(γ , γ )γ = 0 by ∂t 0 antisymmetry of R, the vector field J0(t) = γ (t) satisfies the Jacobi equation 0 0 0 with J0(0) = γ (0),J0(0) = 0. Similarly, J1(t) = tγ (t) is a Jacobi field 0 0 satisfying J1(0) = 0,J1(0) = γ (0). It is easy to see that J0 is the variation field of the variation F (t, u) = γ(t + u), and J1 is the variation field of the variation F (t, u) = γ(teu). Therefore, these two Jacobi fields just reflect the possible reparametrizations of γ, and don’t tell us anything about the behavior of geodesics other than γ itself. 5 To distinguish these two trivial cases from more informative ones, we make the following definitions.
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