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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
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