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File: Geometry Pdf 167039 | Dg Item Download 2023-01-25 01-12-02
sissa dierential geometry boris dubrovin contents 1 geometry of manifolds 3 1 1 denition of smooth manifolds 3 1 2 tangent space to a manifold 8 1 3 vector elds ...

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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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