PROBLEM SET 4: JACOBI FIELD DUE: JUNE 4, 2024 Instruction: Please work on at least Five problems of your interest, and in each problem you are supposed to work on at least TWO sub-problems. 1. [Jacobi fields for manifolds with constant sectional curvature along γ] Let (M, g) be a Riemannian manifold, and γ : [0, l] → M a normal geodesic, where we assume l ̸= kπ/√ κ if κ > 0. Suppose M has constant sectional curvature κ along γ, i.e. i.e. K(Πγ(t) ) = κ for any 2-dimensional plane Πγ(t) ∋ γ˙(t). (a) Prove: The Jacobi field V along γ with V (0) = 0 and V (l) = Xl ∈ ( ˙γ(l))⊥ is V (t) = snκ(t) snκ(l) X(t), where X is the parallel vector field along γ with X(l) = Xl . (b) Given any X0 ∈ Tγ(0)M and Xl ∈ Tγ(l)M, find the Jacobi field with V (0) = X0, V (l) = Xl . 2. [Characterizing constant curvature via Jacobi field] We say a vector field Y along γ is ✿✿✿✿✿✿✿ almost ✿✿✿✿✿✿✿✿ parallel if there is a smooth function f so that Y (t) = f(t)X(t), where X is parallel along γ. As we have seen, any normal Jacobi field along a geodesic on a constant curvature space is almost parallel. (a) Suppose γ : [0, l] → M is a geodesic in (M, g) with γ(0) = p, so that any normal Jacobi field along γ is almost parallel. Let V ⊂ TpM, and V0 a small neighborhood of 0 in V such that expp : V0 → N = expp (V0) is a diffeomorphism. Suppose γ([0, l]) ⊂ N. Prove: P γ 0,l(V ) = Tγ(l)N. (b) Suppose for any geodesic γ with γ(0) = p, any normal Jacobi field along γ is almost parallel. Prove: for any pairwise orthogonal vectors u, v, w ∈ TpM, ⟨R(u, v)u, w⟩ = 0. (c) Suppose m ≥ 3. Prove: If any normal Jacobi field along any geodesic in M is almost parallel, then M has constant sectional curvature. 3. [Square of distance in normal coordinates] Consider two geodesics γ1(t) = expp (tv) and γ2(t) = expp (tw) emanating from p. Let g(s) = d 2 (γ1(s), γ2(s)). To estimate L(s) for s small, consider the variation f(t, s) = σs(t) := expγ1(s) (t exp−1 γ1(s) (γ2(s))). Let ft(t, s) = dft,s(∂/∂t) and fs(t, s) = dft,s(∂/∂s) as usual. Note that for s small, σs is the minimizing geodesic from γ1(s) to γ2(s), and g(s) = ∥σ˙ s(t)∥ 2 = ∥ft(t, s)∥ 2 . [In what follows, although we use usual notation for the connection, they should be understood as the induced connection.] (a) Show that ft(t, 0) = 0, fs(0, s) = ˙γ1(s), fs(1, s) = ˙γ2(s). (b) Show that (∇∂/∂tft)(t,s) = 0, (∇∂/∂sfs)(0,s) = 0, (∇∂/∂sfs)(1,s) = 0. (c) Show that for each fixed s, fs(t, s) is a Jacobi field along γs, and (∇∂/∂t∇∂/∂tfs)(t,0) = 0. Conclude that fs(t, 0) is linear and thus fs(t, 0) = v + t(w − v). (d) Show that g ′ (0) = 0, g′′(0) = 2|v − w| 2 . 1
2 PROBLEM SET 4: JACOBI FIELD DUE: JUNE 4, 2024 (e) Show that (∇∂/∂t∇∂/∂t∇∂/∂sfs)(t,0) = 0, which implies (∇∂/∂sfs)(t,0) is linear in t. Conclude that (∇∂/∂sfs)(t,0) = 0, (∇∂/∂t∇∂/∂sfs)(t,0) = 0, (∇∂/∂s∇∂/∂sft)(t,0) = 0, and g ′′′(0) = 0. (f) Show that g ′′′′(0) = 8Rm(v, w, v, w). (g) Conclude that g(s) = |v − w| 2 s 2 + 1 3Rm(v, w, v, w)s 4 + O(s 5 ) and thus d(γ1(s), γ2(s)) = |v − w|s + 1 6 Rm(v, w, v, w) |v − w| s 3 + O(s 4 ). 4. [Expansion of metric in normal coordinates: next term] (a) Prove: In a Riemannian normal coordinates, near p we have gij = δij − 1 3 Rikjlx kx l − 1 6 Rikjl;rx kx lx r + O(|x| 4 ). What can you say about the coefficients of higher order terms? (b) Expand det(gij ) up to order 3 near p. (c) Prove Bianchi Identity II using normal coordinates (d) Prove: A chart is a Riemannian normal coordinate system if and only if for any i, gijx j = x i . 5. [Locally symmetric space] Let (M, g) be a locally symmetric space, i.e. if ∇XR = 0 for all X ∈ Γ∞(TM). Let γ : [0, a] → M be a geodesic in M with p = γ(0), Xp = ˙γ(0). (a) Let X, Y, Z be vector fields that are parallel along γ. Prove: R(X, Y )Z is also parallel along γ. (b) Define a linear transformation KXp : TpM → TpM by KXp (Yp) = R(Xp, Yp)Xp. Prove: KXp is self-adjoint. (c) Let λ1, · · · , λm be eigenvalues of KXp , with corresponding eigenvectors e1, · · · , em ∈ TpM. Let ej (t) be the parallel transport of ej along γ. Prove: for all t ∈ [0, ∞), we have Kγ˙ (t) (ei(t)) = λiei(t), i = 1, · · · , m. (d) Let X(t) = Xi (t)ei(t) be a Jacobi field along γ. Show that the Jacobi equation becomes X¨i (t) + λiXi = 0, i = 1, · · · , m. (e) Conclude that the conjugate points of p along γ are given by γ( πk √ λi ), k = 1, 2, · · · , where λi ’s are positive eigenvalues of KXp . 6. [More on cut locus] Let (M, g) be complete. (a) Prove: The function f : SM → R ∪ {∞} defined by f(Xp) = t0, if γp,Xp (t0) is the cut point of p along γ, +∞, if p has no cut point along γp,Xp . is continuous. (b) For any p ∈ M, Cut(p) is closed. (c) M is compact if and only if Cut(p) is nonempty and compact for any p ∈ M
PROBLEM SET 4: JACOBI FIELD DUE: JUNE 4, 2024 3 (d) Let Σ(p) = {tXp | Xp ∈ SpM, 0 ≤ t < f(Xp)}. Prove: expp : Σ(p) → expp (Σ(p)) is a diffeomorphism. (e) Prove: M = expp (Σ(p)) ∪ Cut(p) and expp (Σ(p)) ∩ Cut(p) = ∅. (f) We call a point q ∈ M a regular cut point if there exists at least two minimal geodesic from p to q. Prove: The set of regular cut points is a dense subset of Cut(p). 7. [Smoothness of distance function] For any p ∈ M, consider the distance square function f(q) = 1 2 dist(p, q) 2 . (a) For (M, g) = (S m, gSm) the standard sphere, is f a smooth function? (b) Argue that f is smooth on M \Cut(p). (c) For any q ∈ M \Cut(p), find (∇f)(q). (d) For any q ∈ M \ Cut(p) and Yq ∈ TqM, let X be a Jacobi field along γ q , the minimal geodesic from p to q, so that X(0) = 0, X(dist(p, q)) = Yq. Prove: ∇2f(Yq, Yq) = dist(p, q)⟨∇γ˙ q(dist(p,q))X, X(dist(p, q))⟩. (e) Prove: f is not C 1 function at regular cut points. (f) Can f be everywhere smooth on M if M is compact? 8. [Convex Functions on Riemannian Manifolds] • Let (M, g) be a Riemannian manifold. A function f : M → R is said to be a convex function if for any geodesic γ : [a, b] → M, the function f ◦ γ : [a, b] → R is convex. (a) Prove: If f is a convex function on M, then for any c ∈ R, the sublevel set Mc = {p ∈ M | f(p) < c} is a totally convex subset of M. (b) Prove: If f is smooth, then f is convex if and only if its Hessian ∇2f is positive semidefinite. (c) Let p ∈ M be an arbitrary point, and dp(q) = dist(p, q) is the distance function from p. Prove: There exists an neighborhood U of p so that the distance square function d 2 p is convex on (U, g). • Now suppose (M, g) is a complete simply-connected Riemannian manifold with non-positive sectional curvature. (d) Prove: the distance square function d 2 : M × M → R, d2 (p, q) = [dist(p, q)]2 is convex on M × M. (e) Conclude that for any p ∈ M, the function d 2 p is a convex function on M