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Elementary Differential Geometry

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Elementary Differential Geometry ELEMENTARY DIFFERENTIAL GEOMETRY YONG-GEUN OH { Based on the lecture note of Math 621-2020 in POSTECH { Contents Part 1. Riemannian Geometry 2 1. Parallelism and Ehresman connection 2 2. Affine connections on vector bundles 4 2.1. Local expression of covariant derivatives 6 2.2. Affine connection recovers Ehresmann connection 7 2.3. Curvature 9 2.4. Metrics and Euclidean connections 9 3. Riemannian metrics and Levi-Civita connection 10 3.1. Examples of Riemannian manifolds 12 3.2. Covariant derivative along the curve 13 4. Riemann curvature tensor 15 5. Raising and lowering indices and contractions 17 6. Geodesics and exponential maps 19 7. First variation of arc-length 22 8. Geodesic normal coordinates and geodesic balls 25 9. Hopf-Rinow Theorem 31 10. Classification of constant curvature surfaces 33 11. Second variation of energy 34 Part 2. Symplectic Geometry 39 12. Geometry of cotangent bundles 39 13. Poisson manifolds and Schouten-Nijenhuis bracket 42 13.1. Poisson tensor and Jacobi identity 43 13.2. Lie-Poisson space 44 14. Symplectic forms and the Jacobi identity 45 15. Proof of Darboux' Theorem 47 15.1. Symplectic linear algebra 47 15.2. Moser's deformation method 48 16. Hamiltonian vector fields and diffeomorhpisms 50 17. Autonomous Hamiltonians and conservation law 53 18. Completely integrable systems and action-angle variables 55 18.1. Construction of angle coordinates 56 18.2. Construction of action coordinates 57 18.3. Underlying geometry of the Hamilton-Jacobi method 61 19. Lie groups and Lie algebras 62 1 2 YONG-GEUN OH 20. Group actions and adjoint representations 67 21. Lie Poisson space g∗ and the coadjoint action of G 69 22. Symplectic action and the moment map 73 23. Marsden-Weinstein symplectic reduction theorem 78 24. Functorial properties of moment map 82 References 84 Part 1. Riemannian Geometry 1. Parallelism and Ehresman connection For a given vector bundle π : E ! M, we have natural exact sequence deπ 0 ! ker deπ ! TeE ! Tπ(e)M ! 0: The kernel ker dπ ⊂ TE forms a subbunle of rank k = rank E which defines an integrable distribution on the manifold E. Its integral submanifolds are nothing −1 but the fibers π (x), x 2 M. We denote VTeE = ker deπ and call it as the vertical tangent space of E at e, and the union [ VTE = VTeE =: VeE e2E the vertical subbundle. There is no canonical notion of horizontal subspaces. Definition 1.1. An Ehresmann connection is an assignment of subspaces He ⊂ TeE complementary to VeE in TeE at each e 2 E such that (1) HE := [e2EHe is a subbundle of TE such that TeE = HeE ⊕ VeE, (2) For any e 2 E and λ 2 R, Hλe = dmλ(He) where mλ : E ! E is the scalar multiplication by λ. ∼ Statement (2) in particular implies H0x = T0x 0E for the zero section 0E = M and 0x 2 0E is the point corresponding to x 2 M. Postponing the discussion on the existence of Ehresmann connection till later, we proceed. Denote an Ehresmann connection of E as a splitting Γ: TE = HE ⊕ VE for HE ⊂ TE the aforementioned subbundle. We now recall the definition of the pull-back bundle f ∗E for a smooth map f : N ! M: f ∗E := f(n; e) 2 N × E j f(n) = π(e)g: Denote by [n; e] the element of f ∗E represented by (n; e) 2 N × E. Then we can express ∗ T[n;e](f E) = f(v; ξ) 2 TnN × TeE j df(v) = dπ(ξ)g: Definition 1.2 (Pull-back connection). Let π : E ! M be a vector bundle and f : N ! M be a smooth map. The pull-back connection f ∗Γ is given by the choice of a horizontal subbunle of T (f ∗E) given by ∗ ∼ H[n;e](f E) := f(v; ξ) 2 TnN × TeE j df(v) = dπ(ξ); ξ 2 Hf(n)Eg = Hf(n)E: ELEMENTARY DIFFERENTIAL GEOMETRY 3 Let π : E ! M be a the vector bundle of rank k. Consider the pull-back bundle γ∗E ! I = [0; 1] for the given path γ : I ! M. Let me start with more abstract terms. The parallel transport has the following interpretation. Let F = γ∗E ! [0; 1] = B be the above pull-back bundle and TF = H ⊕ V its splitting induced by the pull-back connection. Note that H is a one-dimensional distribution on F and that the projection dπ : H ! TB induces @ an isomorphism that TB = T [0; 1] carries a canonical global frame f @t g. Therefore this lifts to a smooth (autonomous) vector field X on F = γ∗E (regarded as a manifold) given by @ X(e) := (d πj )−1 e H @t on F . By definition @ deπ(X(e)) = : @t t Therefore we can use t itself as the time for the ODEe _ = X(e) on F . ∗ We consider the integral curve γe of X issued at e 2 γ E, i.e., satisfying the ODE dγ e = X(γ(t)); γ(0) = e: dt e e This exists by the existence theorem of the first-order ODE on whole [0; 1] and is smooth by the smooth dependence of the solution of ODE on its initial data. ∗ By denoting the solution by γee, we have γee(t) 2 γ Ejt and interpret is as a curve in E noting that ∗ γ Ejt = Eγ(t) Definition 1.3. The parallel transport Πγ : Eγ(0) ! Eγ(1) is defined by Πγ (e) := γee(1): Proposition 1.4. The map Πγ : Eγ(0) ! Eγ(1)) is a linear map. Proof. We start with proving Πγ (λe) = λΠγ (e). Let γee is the horizontal lift of γ with γee(0) = e. We consider the curve ` : [0; 1] ! E defined by `(t) = λγee(t). Clearly `(t) = λe and dγee j 2 H since γ is the horizontal lift of γ. We compute dt t γee(t) ee the derivative d dγ _ ee `(t) = (λγe) = dmλ 2 Hλγe(t) dt e t dt t e by the defining property of the horizontal subspace in the definition of Ehresmann connection. It is also a lift of γ. Therefore by the uniqueness of the lift, we must have λγee(t) = γeλe(t) for all t 2 [0; 1]. In particular, we have λγee(1) = γeλe(1). By definition, the last is equivalent to Πγ (λe) = λΠγ (e). Next we quote the following lemma Lemma 1.5. Let V , W be vector spaces and f : V ! W be a map differentiable at 0 2 V . If f satisfies f(λv) = λf(v) for all λ 2 R and v 2 V . Then f = df(0). In particular f is a linear map. 4 YONG-GEUN OH Proof. First by setting λ = 0, we obtain f(0) = 0. Therefore for λ 6= 0, we have 1 1 f(v) = f(cv) = (f(cv) − f(0)): c c By letting c ! 0, we obtain 1 lim (f(cv) − f(0)) = df(0)(v) c!0 c since f is assumed to be differentiable at 0. This finishes the proof. Combining the above, we have proved the proposition. Now we introduce the notion of covariant derivative. Definition 1.6. Let Γ : TE = HE ⊕ VE be an Ehresmann connection. Let s 2 Γ(E) be a section of E. Then we define the covariant derivative rvs of s along v 2 TxM to be d t −1 rvs := (Πγ j0) (s(γ(t))) dt t=0 t for a (and so any) germ of curves γ :(−, ) ! M, where Πγ j0 is the parallel transport along γ from 0 to t. It can be checked that this definition does not depend on the choice of γ satisfying γ0(0) = v. Remark 1.7. We can also write v ∼ rvs = ΠΓ(ds(v)) 2 VTs(x) = Ex i.e., `the covariant derivative is the vertical projection of ds(v) 2 Ts(x)E with respect ∼ to the Ehresmann connection Γ : TE ⊕ HE ⊕ VE after identification of VTs(x) = Ex." The following properties can be also checked easily. Lemma 1.8. Let x 2 M. The assignment s 2 Γ(E) ! Ex satisfies (1) rv1+v2 s = rv1 s + rv2 s for v1; v2 2 TxM, (2) rcvs = crcs for all c 2 R and v 2 TxM, 1 (3) rv(fs) = f(x)rvs + v[f]rvs for any f 2 C (M), where v[f] is the direc- tional derivative of f along v at x, (4) rv(s1 + s2) = rvs1 + rvs2 for si 2 Γ(E) and v 2 TxM. We would like to mention that when E is the trivial line bundle M × R, rv is the same as the tangent vector v as a derivation at x with values in R. 2. Affine connections on vector bundles Definition 2.1. Let x 2 M. A linear map D : Γ(E) ! Ex is called defines a derivation at x with values in E if D satisfies the properties (1) - (4) above. A smooth family of such D over M is called a differential operator of order 1 on Γ(E). An example of a differential operator of order 1 on Γ(E) is rX for any vector field X on M. ELEMENTARY DIFFERENTIAL GEOMETRY 5 Definition 2.2 (Affine connection). An affine connection on E ! M is the as- signment of X 7! rX that satisfies (1) rX1+X2 s = rX1 s + rX2 s, (2) rcX s = crX s, 1 (3) rX (fs) = frX s + X[f]rX s for any f 2 C (M), (4) rX (s1 + s2) = rX s1 + rX s2 for si 2 Γ(E).
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