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SISSA Differential Geometry Boris DUBROVIN Contents 1 Geometry of Manifolds 3 1.1 Definition of smooth manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . 3 1.2 Tangent space to a manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3 Vector fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1.4 Smooth functions on manifolds, partitions of unity. . . . . . . . . . . . . . . 22 1.5 Immersions and submersions . . . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.6 Sardtheorem. EmbeddingsofcompactmanifoldsintoEuclideanspaces. Transver- sality. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 2 First examples of topological invariants 43 2.1 Orientation. Topological degree of a smooth map . . . . . . . . . . . . . . . . 43 2.2 Intersection index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.3 Index of a vector field on a manifold . . . . . . . . . . . . . . . . . . . . . . . 53 2.4 Morse index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 2.5 Lefschetz number. Brouwer theorem . . . . . . . . . . . . . . . . . . . . . . . 56 3 Tensors on a manifold. Integration of differential forms. Cohomology 56 3.1 Tensors on manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 3.2 Vector bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.3 Integration of differential forms. Cohomology . . . . . . . . . . . . . . . . . . 62 3.4 Homotopy invariance of cohomologies. Degree of a smooth map and integrals of differential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 4 Riemannian Manifolds 79 4.1 Riemannian metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 1 4.2 Tensors on a Riemannian manifold . . . . . . . . . . . . . . . . . . . . . . . . 86 4.3 Riemannian manifolds as metric spaces . . . . . . . . . . . . . . . . . . . . . . 90 4.4 Approximation theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95 4.5 Isometries of Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . 97 4.6 Affine connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99 4.7 Parallel transport. Curvature of an affine connection . . . . . . . . . . . . . . 104 4.8 The Levi-Civita connection and curvature of Riemannian manifolds . . . . . . 111 4.9 Geodesics on a Riemannian manifold . . . . . . . . . . . . . . . . . . . . . . . 117 4.10 Gaussian connection on surfaces. Curvature of curves and surfaces . . . . . . 127 4.11 Curvature of surfaces in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 4.12 Gauss–Bonnet theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141 4.13 Conformal structures on two-dimensional Riemannian manifolds and Laplace– Beltrami equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 4.14 Geometry of sphere and pseudosphere in conformal coordinates . . . . . . . . 150 4.15 Surfaces of constant curvature. Liouville equation . . . . . . . . . . . . . . . . 154 4.16 Differential geometry versus topology: Gauss–Bonnet formula and Gauss map 159 4.17 Second variation in the theory of geodesics . . . . . . . . . . . . . . . . . . . . 164 4.18 Index theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172 4.19 Lie groups as Riemannian manifolds . . . . . . . . . . . . . . . . . . . . . . . 179 4.20 Differential geometry of complex manifolds . . . . . . . . . . . . . . . . . . . 184 5 Symplectic manifolds. Poisson manifolds 190 5.1 Basic definitions. Poisson brackets . . . . . . . . . . . . . . . . . . . . . . . . 190 5.2 Poisson symmetries. Hamiltonian flows as symplectomorphisms . . . . . . . . 197 5.3 First integrals of Hamiltonian systems . . . . . . . . . . . . . . . . . . . . . . 202 5.4 Darboux Lemma. Casimirs and symplectic leaves on Poisson manifolds . . . . 204 5.5 Poisson cohomology and supermanifolds . . . . . . . . . . . . . . . . . . . . . 208 5.6 Symplectic reduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 5.7 Evolution PDEs as infinite-dimensional Hamiltonian systems . . . . . . . . . 218 5.8 Lagrangian submanifolds, generating functions and Hamilton–Jacobi equation 223 5.9 Symplectic group . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228 5.10 Lagrangian Grassmannian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 5.11 Maslov index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 5.12 Applications to quasiclassical asymptotics of solutions to Schr¨odinger equation 239 2 1 Geometry of Manifolds 1.1 Definition of smooth manifolds Spaces that locally look like Euclidean spaces are called manifolds. Let us give a definition of a smooth manifold. Definition 1.1.1 1) An atlas on a set M is a collection of • subsets Uα ⊂ M that cover all M labeled by an at most numerable set of indices I ∋ α; • for any α ∈ I a one-to-one map ϕ from U to an open domain in the Euclidean space α α Rn is given ϕ :U →ϕ (U )⊂Rn (1.1.1) α α α α The pair (U ,ϕ ) is called a coordinate chart on M. The Euclidean coordinates in Rn α α 1 n n (x ,...,x ) ∈ ϕ (U ) ⊂ R (1.1.2) α α α α define coordinates on the subsets U ⊂ M, i.e., α for P ∈U x1(P),...,xn(P) = ϕ (P). α α α α 2) For any pair of intersecting sets U ∩U 6= ∅ the domains ϕ (U ∩U ) and ϕ (U ∩U ) α β α α β β α β are open in Rn and the one-to-one map −1 ϕ ◦ϕ :ϕ (U ∩U )→ϕ (U ∩U ) (1.1.3) β α α α β β α β is smooth. Since the inverse map −1 ϕ ◦ϕ :ϕ (U ∩U )→ϕ (U ∩U ) α β β α β α α β is smooth as well, we conclude that the transition maps (1.1.3) are all diffeomorphisms. 3) A subset V ⊂ M is called open if its intersections with coordinate charts ϕ (V ∩U )⊂Rn α α are open for all α ∈ I. This definition provides a structure of topological space on M. Aset M equipped with an atlas of coordinate charts with smooth transition maps is called a smooth manifold of dimension n if it is a Hausdorff second countable topological space. Recall that a topological space X is called Hausdorff if, for any pair of distinct points P, Q ∈ X there exist disjoint open neighborhoods U ∋ P, V ∋ Q, U ∩ V = ∅. It is called second countable if one can find a countable collection B of open subsets of X such that any open U ⊂ X is a union of subsets from B. 3 Figure 1: Transition maps on a smooth manifold Counterexamples. To construct a “non-Hausdorff manifold” take two copies R± of real line. Denote x the standard coordinates on these lines. Identify the negative points ± x with x on these lines. The resulting set M is covered by two coordinate charts. The − + points 0 and 0 are distinct; their arbitrary open neighborhoods intersect. To construct a + − “non-second countable manifold” one can take a disjoint union of an uncountable number of copies of real line. Example 1.1.2 The n-dimensional Euclidean space itself, or also any open domain in it, are examples of smooth manifolds. Example 1.1.3 The unit sphere Sn ⊂ Rn+1 is an example of a n-dimensional manifold covered with two coordinate charts. The maps π± can be described as stereographic projections of the sphere from the poles P± = (0,0,...,±1) π :Sn\P →Rn + + 1 n 1 n+1 x x 1 n π+(x ,...,x ) = , . . . , =: (x ,...,x ) n+1 n+1 + + 1−x 1−x (1.1.4) π− : Sn \ P− → Rn 1 n 1 n+1 x x 1 n π−(x ,...,x ) = , . . . , =: (x ,...,x ) n+1 n+1 − − 1+x 1+x 4
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