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Theorem 1.2.1. (Viterbi) Every hypersurface of contact type in \( {\mathbb{R}}^{2n} \) has a closed characteristic. | Proof: See Theorem 12.4.6 on page 486. | No |
Theorem 1.2.3 (Nonsqueezing theorem) If there exists a symplectic embedding \( {B}^{2n}\left( r\right) \hookrightarrow {Z}^{2n}\left( R\right) \), then \( r \leq R \) . | Proof: The result is restated in Theorem 12.1.1 and proved on page 484. | No |
Lemma 2.1.1 For any subspace \( W \subset V \) ,\n\n\[ \n\dim W + \dim {W}^{\omega } = \dim V,\;{W}^{\omega \omega } = W.\n\] | Proof: Define a map \( {\iota }_{\omega } \) from \( V \) to the dual space \( {V}^{ * } \) by setting\n\n\[ \n{\iota }_{\omega }\left( v\right) \left( w\right) = \omega \left( {v, w}\right) .\n\]\n\nSince \( \omega \) is nondegenerate \( {\iota }_{\omega } \) is an isomorphism. It identifies \( {W}^{\omega } \) with t... | No |
Theorem 2.1.3 Let \( \\left( {V,\\omega }\\right) \) be a symplectic vector space of dimension \( {2n} \) . Then there exists a basis \( {u}_{1},\\ldots ,{u}_{n},{v}_{1},\\ldots ,{v}_{n} \) such that\n\n\[ \n\\omega \\left( {{u}_{j},{u}_{k}}\\right) = \\omega \\left( {{v}_{j},{v}_{k}}\\right) = 0,\\;\\omega \\left( {{u... | Proof: The proof is by induction over \( n \) . Since \( \\omega \) is nondegenerate there exist vectors \( {u}_{1},{v}_{1} \\in V \) such that \( \\omega \\left( {{u}_{1},{v}_{1}}\\right) = 1 \) . Hence the subspace spanned by \( {u}_{1} \) and \( {v}_{1} \) is symplectic. Let \( W \) denote its symplectic complement.... | Yes |
Corollary 2.1.4 Let \( V \) be a 2n-dimensional real vector space and let \( \omega \) be a skew-symmetric bilinear form on \( V \) . Then \( \omega \) is nondegenerate if and only if its \( n \) -fold exterior power is nonzero, i.e. \( {\omega }^{n} = \omega \land \cdots \land \omega \neq 0 \) . | Proof: Assume first that \( \omega \) is degenerate. Let \( v \neq 0 \) such that \( \omega \left( {v, w}\right) = 0 \) for all \( w \in V \) . Now choose a basis \( {v}_{1},\ldots ,{v}_{2n} \) of \( V \) such that \( {v}_{1} = v \) . Then \( {\omega }^{n}\left( {{v}_{1},\ldots ,{v}_{2n}}\right) = 0 \) . Conversely, su... | Yes |
Lemma 2.1.5 Every isotropic subspace of \( V \) is contained in a Lagrangian subspace. Moreover, every basis \( {u}_{1},\ldots ,{u}_{n} \) of a Lagrangian subspace \( \Lambda \) can be extended to a symplectic basis of \( \left( {V,\omega }\right) \) . | Proof: Let \( W \) be an isotropic subspace, i.e. \( W \subset {W}^{\omega } \) . If the subspace \( {W}_{1} \) is obtained by adjoining some vector \( v \in {W}^{\omega } \smallsetminus W \) to \( W \), then \( \omega \) vanishes on \( {W}_{1} \) . Hence a maximal isotropic subspace must satisfy \( W = {W}^{\omega } \... | Yes |
Lemma 2.1.7 Let \( \left( {V,\omega }\right) \) be a symplectic vector space and let \( W \subset V \) be a coisotropic subspace. Then the following holds. (i) The quotient\n\n\[ \overline{W} \mathrel{\text{:=}} W/{W}^{\omega } \]\n\ncarries a unique symplectic structure \( \bar{\omega } \) such that the restriction \(... | Proof: Denote \( \left\lbrack w\right\rbrack \mathrel{\text{:=}} w + {W}^{\omega } \in \bar{W} \) for \( w \in W \) . By the definition of coisotropic, \( {W}^{\omega } \) is an isotropic subspace of \( W \) and \( \omega \left( {v, w}\right) = 0 \) for \( v \in {W}^{\omega } \) and \( w \in W \) . Hence \( \omega \lef... | Yes |
Lemma 2.2.2 Let \( \Psi \in \operatorname{Sp}\left( {2n}\right) \) . Then\n\n\[ \lambda \in \sigma \left( \Psi \right) \; \Leftrightarrow \;{\lambda }^{-1} \in \sigma \left( \Psi \right) \]\n\nand the multiplicities of \( \lambda \) and \( {\lambda }^{-1} \) agree. If \( \pm 1 \) is an eigenvalue of \( \Psi \) then it ... | Proof: The first statement follows from the fact that \( {\Psi }^{\mathrm{T}} \) is similar to \( {\Psi }^{-1} \) :\n\n\[ {\Psi }^{\mathrm{T}} = {J}_{0}{\Psi }^{-1}{J}_{0}^{-1}. \]\n\nHence the total multiplicity of all eigenvalues not equal to 1 or -1 is even. Since the determinant is the product of all eigenvalues it... | Yes |
Lemma 2.2.3 If \( P = {P}^{\mathrm{T}} \in \operatorname{Sp}\left( {2n}\right) \) is a symmetric, positive definite symplectic matrix then \( {P}^{\alpha } \in \operatorname{Sp}\left( {2n}\right) \) for every real number \( \alpha \geq 0 \) . | Proof: Every positive definite symmetric matrix has positive eigenvalues and an orthonormal basis of eigenvectors. Hence there is an orthogonal decomposition\n\n\[ \n{\mathbb{R}}^{2n} = {\bigoplus }_{i = 1}^{k}{E}_{i},\;{E}_{i} = \ker \left( {{\lambda }_{i}\mathbb{1} - P}\right) , \n\]\n\nwhere \( {\lambda }_{i} > 0, i... | Yes |
Proposition 2.2.4 (i) The unitary group \( \mathrm{U}\left( n\right) \) is a maximal compact subgroup of the symplectic linear group \( \operatorname{Sp}\left( {2n}\right) \) . | Proof: We prove (i). Suppose \( \mathrm{G} \) is a compact subgroup of \( \mathrm{{Sp}}\left( {2n}\right) \) that contains \( \mathrm{U}\left( n\right) \) but is not equal to \( \mathrm{U}\left( n\right) \) . Then there is an element \( \Psi \in \mathrm{G} \smallsetminus \mathrm{U}\left( n\right) \) . We saw above that... | Yes |
Proposition 2.2.6 The fundamental group of \( \mathrm{U}\left( n\right) \) is isomorphic to the integers. The complex determinant map \( \mathop{\det }\limits_{\mathbb{C}} : \mathrm{U}\left( n\right) \rightarrow {S}^{1} \) induces an isomorphism of fundamental groups. | Proof: The determinant map \( \mathop{\det }\limits_{\mathbb{C}} : \mathrm{U}\left( n\right) \rightarrow {S}^{1} \) is a fibration with fibre \( \mathrm{{SU}}\left( n\right) \) . Hence the homotopy exact sequence\n\n\[ \n{\pi }_{1}\left( {\mathrm{{SU}}\left( n\right) }\right) \rightarrow {\pi }_{1}\left( {\mathrm{U}\le... | Yes |
Theorem 2.2.12 There exists a unique function \( \mu \), called the Maslov index, which assigns an integer \( \mu \left( \Psi \right) \) to every loop\n\n\[ \Psi : \mathbb{R}/\mathbb{Z} \rightarrow \operatorname{Sp}\left( {2n}\right) \]\n\nof symplectic matrices and satisfies the following axioms:\n\n(homotopy) Two loo... | Proof: Define the map \( \rho : \operatorname{Sp}\left( {2n}\right) \rightarrow {S}^{1} \) by\n\n\[ \rho \left( \Psi \right) \mathrel{\text{:=}} \mathop{\det }\limits_{\mathbb{C}}\left( {X + {iY}}\right) ,\;\left( \begin{matrix} X - Y \\ Y\;X \end{matrix}\right) \mathrel{\text{:=}} \Psi {\left( {\Psi }^{\mathrm{T}}\Psi... | Yes |
Lemma 2.3.1 Let \( X \) and \( Y \) be real \( n \times n \) matrices and define \( \Lambda \subset {\mathbb{R}}^{2n} \) by\n\n\[ \n\Lambda = \operatorname{im}Z,\;Z = \left( \begin{array}{l} X \\ Y \end{array}\right) .\n\]\n\nThen \( \Lambda \in \mathcal{L}\left( n\right) \) if and only if the matrix \( Z \) has rank \... | Proof: Given two vectors \( z = \left( {{Xu},{Yu}}\right) \) and \( {z}^{\prime } = \left( {X{u}^{\prime }, Y{u}^{\prime }}\right) \) in \( \Lambda \), we have by (1.1.21) that \( {\omega }_{0}\left( {z,{z}^{\prime }}\right) = {u}^{\mathrm{T}}\left( {{X}^{\mathrm{T}}Y - {Y}^{\mathrm{T}}X}\right) {u}^{\prime } \) . This... | Yes |
Lemma 2.3.2 (i) If \( \Lambda \in \mathcal{L}\left( n\right) \) and \( \Psi \in \operatorname{Sp}\left( {2n}\right) \) then \( {\Psi \Lambda } \in \mathcal{L}\left( n\right) \) . | Statement (i) is obvious. | No |
Theorem 2.3.7 There exists a function \( \mu \), called the Maslov index, which assigns an integer \( \mu \left( \Lambda \right) \) to every loop \( \Lambda : \mathbb{R}/\mathbb{Z} \rightarrow \mathcal{L}\left( n\right) \) of Lagrangian subspaces and satisfies the following axioms:\n\n(homotopy) Two loops in \( \mathca... | Proof: Define the map \( \rho : \mathcal{L}\left( n\right) \rightarrow {S}^{1} \) by \( {}^{9} \)\n\n\[ \rho \left( \Lambda \right) \mathrel{\text{:=}} {\det }_{\mathbb{C}}\left( {U}^{2}\right) ,\;\operatorname{im}\left( \begin{array}{l} X \\ Y \end{array}\right) = \Lambda ,\;U \mathrel{\text{:=}} X + {iY} \in \mathrm{... | Yes |
Theorem 2.4.1 (Affine nonsqueezing) Let \( \psi \in \operatorname{ASp}\left( {2n}\right) \) be an affine sym-plectomorphism and assume that \( \psi \left( {{B}^{2n}\left( r\right) }\right) \subset {Z}^{2n}\left( R\right) \) . Then \( r \leq R \) . | Proof: Assume without loss of generality that \( r = 1 \) and write \( \psi \left( z\right) = {\Psi z} + {z}_{0} \) . Define\n\n\[ u \mathrel{\text{:=}} {\Psi }^{\mathrm{T}}{e}_{1},\;v \mathrel{\text{:=}} {\Psi }^{\mathrm{T}}{f}_{1},\;a \mathrel{\text{:=}} \left\langle {{e}_{1},{z}_{0}}\right\rangle ,\;b \mathrel{\text... | Yes |
Theorem 2.4.2 (Affine rigidity) Let \( \Psi \in {\mathbb{R}}^{{2n} \times {2n}} \) be a nonsingular matrix such that \( \Psi \) and \( {\Psi }^{-1} \) have the linear nonsqueezing property. Then \( \Psi \) is either symplectic or anti-symplectic. | Proof: Assume that \( \Psi \) is neither symplectic nor anti-symplectic. Then neither is \( {\Psi }^{\mathrm{T}} \) and so there exist vectors \( u, v \in {\mathbb{R}}^{2n} \) such that\n\n\[{\omega }_{0}\left( {{\Psi }^{\mathrm{T}}u,{\Psi }^{\mathrm{T}}v}\right) \neq \pm {\omega }_{0}\left( {u, v}\right)\]\n\nPerturbi... | Yes |
Theorem 2.4.4 Let \( \Psi : {\mathbb{R}}^{2n} \rightarrow {\mathbb{R}}^{2n} \) be a linear map. Then the following are equivalent.\n\n(i) \( \Psi \) preserves the linear symplectic width of ellipsoids centred at zero.\n\n(ii) The matrix \( \Psi \) is either symplectic or anti-symplectic, i.e. \( {\Psi }^{ * }{\omega }_... | Proof: It follows from Theorem 2.4.1 that (ii) implies (i). Hence assume (i). We prove that \( \Psi \) has the nonsqueezing property. To see this let \( B \) be a linear symplectic ball of radius \( r \) and \( Z \) be a linear symplectic cylinder of radius \( R \) such that\n\n\( {\Psi B} \subset Z \) .\n\nThen \( {w}... | Yes |
Lemma 2.4.5 Let \( \left( {V,\omega }\right) \) be a symplectic vector space and \( g : V \times V \rightarrow \mathbb{R} \) be an inner product. Then there exists a basis \( {u}_{1},\ldots ,{u}_{n},{v}_{1},\ldots ,{v}_{n} \) of \( V \) which is both \( g \) -orthogonal and \( \omega \) -standard. Moreover, this basis ... | Proof: Consider the vector space \( V = {\mathbb{R}}^{2n} \) with the standard inner product \( g = \langle \cdot , \cdot \rangle \) and assume that\n\n\[ \omega \left( {z, w}\right) = \langle z,{Aw}\rangle \]\n\n is a nondegenerate skew form. Then \( A \) is nondegenerate and \( {A}^{\mathrm{T}} = - A \) . Hence \( {i... | Yes |
Lemma 2.4.6 Given any compact ellipsoid\n\n\\[ E = \\left\\{ {w \\in {\\mathbb{R}}^{2n} \\mid \\mathop{\\sum }\\limits_{{i, j = 1}}^{{2n}}{a}_{ij}{w}_{i}{w}_{j} \\leq 1}\\right\\} ,\\]\n\nthere is a symplectic linear transformation \\( \\Psi \\in \\operatorname{Sp}\\left( {2n}\\right) \\) such that \\( {\\Psi E} = E\\l... | Proof: Consider the inner product\n\n\\[ g\\left( {v, w}\\right) = \\mathop{\\sum }\\limits_{{i, j = 1}}^{{2n}}{a}_{ij}{v}_{i}{w}_{j} \\]\n\non \\( {\\mathbb{R}}^{2n} \\) . Then the ellipsoid \\( E \\) is given by\n\n\\[ E = \\left\\{ {w \\in {\\mathbb{R}}^{2n} \\mid g\\left( {w, w}\\right) \\leq 1}\\right\\} . \\]\n\n... | Yes |
Theorem 2.4.8 Let \( E \subset {\mathbb{R}}^{2n} \) be an ellipsoid centred at 0 . Then\n\n\[ \n{w}_{L}\left( E\right) = \mathop{\sup }\limits_{{B \subset E}}{w}_{L}\left( B\right) = \mathop{\inf }\limits_{{Z \supset E}}{w}_{L}\left( Z\right) ,\n\]\n\nwhere the supremum runs over all affine symplectic balls contained i... | Proof: Assume that \( E \) has symplectic spectrum \( 0 < {r}_{1} \leq {r}_{2} \leq \cdots \leq {r}_{n} \) . Then there exists a symplectic matrix \( \Psi \in \operatorname{Sp}\left( {2n}\right) \) such that\n\n\[ \n{\Psi E} = E\left( {{r}_{1},\ldots ,{r}_{n}}\right) .\n\]\n\nHence\n\n\[ \n{\Psi }^{-1}{B}^{2n}\left( {r... | Yes |
Proposition 2.5.1 Let \( V \) be a 2n-dimensional real vector space and let \( J \) be a linear complex structure on \( V \) . Then there exists a vector space isomorphism \( \Phi : {\mathbb{R}}^{2n} \rightarrow V \) such that \( {J\Phi } = \Phi {J}_{0} \) . | Proof 1: Every complex vector space has a complex basis. For \( \left( {V, J}\right) \) this means that there exist vectors \( {v}_{1},\ldots ,{v}_{n} \in V \) such that the vectors\n\n\[ \n{v}_{1}, J{v}_{1},\ldots ,{v}_{n}, J{v}_{n} \n\]\n\nform a real basis of \( V \) . The required transformation \( \Phi : {\mathbb{... | Yes |
Proposition 2.5.2 (i) The space \( \mathcal{J}\left( {\mathbb{R}}^{2n}\right) \) is diffeomorphic to the homogeneous space \( \mathrm{{GL}}\left( {{2n},\mathbb{R}}\right) /\mathrm{{GL}}\left( {n,\mathbb{C}}\right) \), and so has two connected components. | Proof: Consider the map\n\n\[ \mathrm{{GL}}\left( {{2n},\mathbb{R}}\right) \rightarrow \mathcal{J}\left( {\mathbb{R}}^{2n}\right) : \Psi \mapsto \Psi {J}_{0}{\Psi }^{-1}. \]\n\nBy Proposition 2.5.1 this map is surjective and it descends to the quotient space \( \mathrm{{GL}}\left( {{2n},\mathbb{R}}\right) /\mathrm{{GL}... | Yes |
Proposition 2.5.4 Let \( \left( {V,\omega }\right) \) be a symplectic vector space and \( J \) be a linear complex structure on \( V \) . Then the following are equivalent.\n\n(i) \( J \) is compatible with \( \omega \) .\n\n(ii) \( \left( {V,\omega }\right) \) has a symplectic basis of the form\n\n\[ \n{v}_{1},\ldots ... | Proof: We prove that (i) implies (ii). By (i) the formula (2.5.4) defines an inner product \( {g}_{J} : V \times V \rightarrow \mathbb{R} \) . By Theorem 2.1.3 there exists a Lagrangian subspace \( \Lambda \subset V \) . Choose an orthonormal basis \( {v}_{1},\ldots ,{v}_{n} \) of \( \Lambda \) with respect to \( {g}_{... | Yes |
Lemma 2.5.5 Let \( \\left( {V,\\omega }\\right) \) be a symplectic vector space of dimension \( {2n} \) . Then the space \( \\mathcal{J}\\left( {V,\\omega }\\right) \) of \( \\omega \) -compatible complex structures on \( V \) is diffeomorphic to the space \( \\mathcal{P} \) of symmetric positive definite symplectic \(... | Proof: By Theorem 2.1.3 we may assume \( V = {\\mathbb{R}}^{2n} \) and \( \\omega = {\\omega }_{0} \) . A matrix \( J \\in {\\mathbb{R}}^{{2n} \\times {2n}} \) is a compatible complex structure if and only if\n\n\[ \n{J}^{2} = - \\mathbb{1},\\;{J}^{\\mathrm{T}}{J}_{0}J = {J}_{0},\\;\\left\\langle {v, - {J}_{0}{Jv}}\\ri... | Yes |
Proposition 2.5.6 Let \( V \) be an even-dimensional real vector space. There exists a \( \mathrm{{GL}}\left( V\right) \) -equivariant smooth map\n\n\[ \mathfrak{{Met}}\left( V\right) \times \Omega \left( V\right) \rightarrow \mathcal{J}\left( V\right) : \left( {g,\omega }\right) \mapsto {J}_{g,\omega } \]\n\nwith the ... | Proof: The proof has three steps. The first step explains the construction of the map (2.5.9) satisfying condition (i).\n\nStep 1. Let \( g \in \mathfrak{{Met}}\left( V\right) \) and \( \omega \in \Omega \left( V\right) \) . Define \( {A}_{g,\omega } \in \mathrm{{GL}}\left( V\right) \) by\n\n\[ \omega \left( {v, w}\rig... | No |
For every \( \omega \in \Omega \left( V\right) \) the map \[ \mathcal{J}\left( {V,\omega }\right) \rightarrow \mathfrak{{Met}}\left( V\right) : J \mapsto {g}_{J} \mathrel{\text{:=}} \omega \left( {\cdot, J \cdot }\right) \] is a homotopy equivalence with homotopy inverse \[ \mathfrak{{Met}}\left( V\right) \rightarrow \... | Proof: The composition of the map \( \mathcal{J}\left( {V,\omega }\right) \rightarrow \mathfrak{{Met}}\left( V\right) : J \mapsto {g}_{J} \) with the map \( \mathfrak{{Met}}\left( V\right) \rightarrow \mathcal{J}\left( {V,\omega }\right) : g \mapsto {J}_{g,\omega } \) is the identity, by Proposition 2.5.6 (i). The conv... | Yes |
Proposition 2.5.8 Every compact subgroup \( \mathrm{G} \subset \mathrm{{Sp}}\left( {2n}\right) \) is conjugate to a subgroup of \( \mathrm{U}\left( n\right) \) . | Proof: The proof has three steps.\n\nStep 1. There exists a symmetric positive definite matrix \( S = {S}^{T} \in {\mathbb{R}}^{{2n} \times {2n}} \) such that \( {\Psi }^{T}{S\Psi } = S \) for every \( \Psi \in \mathrm{G} \) .\n\nThe Haar measure on a compact topological group \( \mathrm{G} \) is the unique bounded lin... | Yes |
Lemma 2.5.12 There is a unique bijection \( {\mathcal{S}}_{n} \rightarrow \mathcal{J}\left( {{\mathbb{R}}^{2n},{\omega }_{0}}\right) : Z \mapsto J\left( Z\right) \) such that for all \( Z \in {\mathcal{S}}_{n} \) and \( \Psi \in \operatorname{Sp}\left( {2n}\right) \), \[ J\left( {i\mathbb{1}}\right) = {J}_{0},\;J\left(... | Proof: It follows from Proposition 2.5.4 that the group \( \operatorname{Sp}\left( {2n}\right) \) acts transitively on \( \mathcal{J}\left( {{\mathbb{R}}^{2n},{\omega }_{0}}\right) \) and it follows from Lemma 2.2.1 that the stabilizer subgroup of \( {J}_{0} \) is \( \operatorname{Sp}\left( {2n}\right) \cap \operatorna... | No |
Corollary 2.5.14 Let \( {\omega }_{t} \) be a smooth family of nondegenerate skew-symmetric bilinear forms on \( V \) depending on a real parameter \( t \) . Then there exists a smooth family of isomorphisms \( {\Psi }_{t} : V \rightarrow V \) such that \( {\Psi }_{0} = \mathbb{1} \) and \( {\Psi }_{t}^{ * }{\omega }_{... | Proof: Fix an inner product \( g \in \mathfrak{{Met}}\left( V\right) \) and let \( {A}_{g} : \Omega \left( V\right) \times {\Lambda }^{2}{V}^{ * } \rightarrow \operatorname{End}\left( V\right) \) be the map constructed in Step 1 of the second proof of Proposition 2.5.13. Let \( {\Psi }_{t} \) be the solution of the dif... | Yes |
Exercise 2.6.2 Let \( \\left( {E,\\omega }\\right) \) be a symplectic vector bundle of rank \( {2n} \) over a manifold \( M \) . Prove that \( E \) admits a system of local trivializations such that the transition maps take values in \( \\operatorname{Sp}\\left( {2n}\\right) \\subset \\operatorname{GL}\\left( {{2n},\\m... | Hint: Construct sections which form a symplectic basis in each fibre, for example by using the method in the proof of Theorem 2.1.3. | No |
Theorem 2.6.3 For \( i = 1,2 \) let \( \left( {{E}_{i},{\omega }_{i}}\right) \) be a symplectic vector bundle over a manifold \( M \) and let \( {J}_{i} \) be an \( {\omega }_{i} \) -tame complex structure on \( {E}_{i} \) . Then the symplectic vector bundles \( \left( {{E}_{1},{\omega }_{1}}\right) \) and \( \left( {{... | Proof: See page 82 below for a proof that does not rely on the theory of classifying spaces. | No |
Proposition 2.6.4 Let \( E \rightarrow M \) be a 2n-dimensional vector bundle.\n\n(i) For every symplectic bilinear form \( \omega \) on \( E \) the space \( \mathcal{J}\left( {E,\omega }\right) \) of \( \omega \) -compatible complex structures on \( E \) and the space \( {\mathcal{J}}_{\tau }\left( {E,\omega }\right) ... | Proof: We prove (i). That \( \mathcal{J}\left( {E,\omega }\right) \) is contractible follows directly from the proof of Corollary 2.5.7. Namely the map \( J \mapsto {g}_{J} \mathrel{\text{:=}} \omega \left( {\cdot, J \cdot }\right) \) defines a homotopy equivalence from \( \mathcal{J}\left( {E,\omega }\right) \) to the... | Yes |
Lemma 2.6.6 Let \( E \rightarrow M \) be a vector bundle with Hermitian structure \( \left( {\omega, J, g}\right) \) . Let \( \gamma : \left\lbrack {0,1}\right\rbrack \rightarrow M \) be a smooth curve and let \( {\Phi }_{0} : {\mathbb{R}}^{2n} \rightarrow {E}_{\gamma \left( 0\right) },{\Phi }_{1} : {\mathbb{R}}^{2n} \... | Proof: We first prove that such a trivialization exists on some interval \( 0 \leq t < \varepsilon \) . Choose \( {s}_{j0} \in {E}_{\gamma \left( 0\right) } \) such that\n\n\[{\Phi }_{0}\zeta = \mathop{\sum }\limits_{j}{s}_{j0}{\zeta }_{j}\]\n\nfor \( \zeta \in {\mathbb{R}}^{2n} \) . We must construct \( {2n} \) sectio... | Yes |
Proposition 2.6.7 A Hermitian vector bundle \( E \rightarrow \sum \) over a compact oriented 2-manifold \( \sum \) with nonempty boundary \( \partial \sum \) admits a unitary trivialization. | Proof: The proof of the proposition is by induction over the number\n\n\[ k\left( \sum \right) = {2g}\left( \sum \right) + \ell \left( \sum \right) \]\n\nwhere \( \ell \left( \sum \right) \geq 1 \) is the number of boundary components and \( g\left( \sum \right) \geq 0 \) is the genus. If \( k\left( \sum \right) = 1 \)... | Yes |
Theorem 2.7.1 There exists a unique functor \( {c}_{1} \), called the first Chern number, that assigns an integer\n\n\[ \n{c}_{1}\left( E\right) \in \mathbb{Z} \n\]\n\nto every symplectic vector bundle \( E \) over a closed oriented 2-manifold \( \sum \) and satisfies the following axioms.\n\n(naturality) Two symplecti... | Proof: See page 87 below. | No |
Lemma 2.7.3 Let \( \sum \) be a compact connected oriented 2-manifold with nonempty boundary. A smooth map \( \Psi : \partial \sum \rightarrow \operatorname{Sp}\left( {2n}\right) \) extends to \( \sum \) if and only if\n\n\[ \deg \left( {\rho \circ \Psi }\right) = 0. \] | Proof: First assume that \( \Psi \) extends to a smooth map \( \sum \rightarrow \operatorname{Sp}\left( {2n}\right) \) . Then the composition \( \rho \circ \Psi : \partial \sum \rightarrow {S}^{1} \) extends to a smooth map \( \sum \rightarrow {S}^{1} \) and hence must have degree zero.\n\nNow assume that \( \partial \... | Yes |
Theorem 2.7.4 Every connection 1-form \( A \) on \( P \) satisfies\n\n\[ \n{c}_{1}\left( P\right) = \frac{i}{2\pi }{\int }_{\sum }{F}_{A} \n\]\n\nMoreover, for every 2-form \( \tau \in {\Omega }^{2}\left( {\sum, i\mathbb{R}}\right) \) with \( {\int }_{\sum }\tau = - {2\pi i}{c}_{1}\left( P\right) \) there exists a conn... | Proof: Decompose \( \sum = {\sum }_{1}{ \cup }_{C}{\sum }_{2} \) and choose sections \( {s}_{j} : {\sum }_{j} \rightarrow P \) . These give rise to trivializations \( {\sum }_{j} \times \mathbb{C} \rightarrow L : \left( {z,\zeta }\right) \mapsto {s}_{j}\left( z\right) \zeta \) and, by the discussion after Theorem 2.7.1... | No |
Theorem 2.7.5 If the section \( s : \sum \rightarrow L \) is transverse to the zero section then the first Chern number of \( L \) is given by\n\n\[ \n{c}_{1}\left( L\right) = \mathop{\sum }\limits_{{s\left( z\right) = 0}}\iota \left( {z, s}\right) \n\]\n\n(2.7.1) | Proof: The proof is an exercise with hint. Cut out a small neighbourhood \( U \) of the zero set of \( s \) and use \( s \) to trivialize the bundle \( L \) over the complement \( {\sum }^{\prime } \mathrel{\text{:=}} \sum \smallsetminus U \) . Use a different method to trivialize \( \sum \) over \( U \) and then compa... | No |
Example 3.1.3 The construction in Example 3.1.2 extends to every 2-dimensional oriented submanifold \( \sum \subset {\mathbb{R}}^{3} \), equipped with a normal vector field \( \nu : \sum \rightarrow {S}^{2} \) so that \( \nu \left( x\right) \bot {T}_{x}\sum \) for all \( x \in \sum \) . An example of a symplectic form ... | It is given by\n\n\[ \n{\omega }_{x}\left( {\xi ,\eta }\right) \mathrel{\text{:=}} \langle \nu \left( x\right) ,\xi \times \eta \rangle = \det \left( {\nu \left( x\right) ,\xi ,\eta }\right) \n\] \n\nfor \( x \in \sum \) and \( \xi ,\eta \in {T}_{x}\sum = \nu {\left( x\right) }^{ \bot } . | Yes |
Proposition 3.1.10 Let \( \\left( {M,\\omega }\\right) \) be a symplectic manifold.\n\n(i) Wherever defined, the Hamiltonian flow \( {\\phi }_{H}^{t} \) is a symplectomorphism which is tangent to the level surfaces of \( H \) .\n\n(ii) For every Hamiltonian function \( H : M \\rightarrow \\mathbb{R} \) and every symple... | Proof: Statement (i) was proved in Proposition 3.1.5. Statement (ii) is the identity\n\n\\[ \n\\iota \\left( {X}_{H \\circ \\psi }\\right) \\omega = d\\left( {H \\circ \\psi }\\right) = {\\psi }^{ * }{dH} = {\\psi }^{ * }\\iota \\left( {X}_{H}\\right) \\omega = \\iota \\left( {{\\psi }^{ * }{X}_{H}}\\right) \\omega .\n... | Yes |
Consider the group \( \Gamma = {\mathbb{Z}}^{2} \times {\mathbb{Z}}^{2} \) with the noncommutative group operation\n\n\[ \left( {{j}^{\prime },{k}^{\prime }}\right) \circ \left( {j, k}\right) = \left( {j + {j}^{\prime },{A}_{{j}^{\prime }}k + {k}^{\prime }}\right) ,\;{A}_{j} \mathrel{\text{:=}} \left( \begin{array}{ll}... | Its fundamental group is \( {\pi }_{1}\left( M\right) = \Gamma \) and hence\n\n\[ {H}_{1}\left( {M;\mathbb{Z}}\right) = \Gamma /\left\lbrack {\Gamma ,\Gamma }\right\rbrack \cong \mathbb{Z} \oplus \mathbb{Z} \oplus \mathbb{Z} \]\n\nHere \( \left\lbrack {\Gamma ,\Gamma }\right\rbrack = 0 \oplus 0 \oplus \mathbb{Z} \oplus... | Yes |
Proposition 3.1.18 The 1-form \( {\lambda }_{\text{can }} \in {\Omega }^{1}\left( {{T}^{ * }L}\right) \) is uniquely characterized by the property that\n\n\[ \n{\sigma }^{ * }{\lambda }_{\text{can }} = \sigma \n\]\n\nfor every 1 -form \( \sigma : L \rightarrow {T}^{ * }L \) . | Proof: In the local coordinates \( x \) on \( L \), a 1 -form \( \sigma \) on \( L \) can be written as\n\n\[ \n\sigma = \mathop{\sum }\limits_{{j = 1}}^{n}{a}_{j}\left( x\right) d{x}_{j} \n\]\n\nWhen regarded as a map from \( L \) to \( {T}^{ * }L \), this 1 -form is represented in the local coordinates \( \left( {x, ... | Yes |
Theorem 3.2.2 (Darboux) Every symplectic form \( \omega \) on \( M \) is locally diffeomorphic to the standard form \( {\omega }_{0} \) on \( {\mathbb{R}}^{2n} \) . | Proof: Apply Lemma 3.2.1 to the case where \( Q \) is a single point and use Theorem 2.1.3. | No |
Theorem 3.2.4 (Moser stability) Let \( M \) be a closed manifold (i.e. a compact manifold without boundary) and suppose that \( {\omega }_{t} \) is a smooth family of cohomologous symplectic forms on \( M \) . Then there is a family of diffeomorphisms \( {\psi }_{t} \) of \( M \) such that\n\n\[ \n{\psi }_{0} = \mathrm... | Proof: To apply Moser’s argument we must find a smooth family of 1-forms \( {\sigma }_{t} \) such that\n\n\[ \nd{\sigma }_{t} = \frac{d}{dt}{\omega }_{t} \n\]\n\nSince the \( {\omega }_{t} \) are cohomologous, each form \( {\omega }_{t} - {\omega }_{0} \) is exact and so are the forms\n\n\[ \n{\tau }_{t} = \frac{d}{dt}... | No |
Theorem 3.3.1 (Symplectic ball extension theorem) Let \( \left( {M,\omega }\right) \) be a connected symplectic manifold without boundary.\n\n(i) Let \( \lambda > 0 \) . Then every symplectic embedding \( \psi : B\left( \lambda \right) \rightarrow M \) extends to a symplectic embedding of \( B\left( r\right) \) into \(... | Proof: The proof has four steps.\n\nStep 1. Let \( \left\lbrack {0,1}\right\rbrack \times B\left( \lambda \right) \rightarrow M : \left( {t, z}\right) \mapsto {\psi }_{t}\left( z\right) \) be a smooth map such that \( {\psi }_{t} : B\left( \lambda \right) \rightarrow M \) is a symplectic embedding for every \( t \) . T... | Yes |
Theorem 3.3.2 (Symplectic isotopy extension theorem) Let \( \left( {M,\omega }\right) \) be a compact symplectic manifold and let \( Q \subset M \) be a compact subset that is a neighbourhood deformation retract and satisfies \( {H}^{2}\left( {M, Q;\mathbb{R}}\right) = 0 \) . Let \( {\phi }_{t} : U \rightarrow M \) be ... | Proof: Choose a neighbourhood \( \mathcal{N} \subset U \) of \( Q \) which retracts onto \( Q \) . Then \( {H}^{ * }\left( {\mathcal{N}, Q;\mathbb{R}}\right) = 0 \) and hence it follows from the cohomology exact sequence of the triple \( \left( {Q,\mathcal{N}, M}\right) \) that \( {H}^{2}\left( {M,\mathcal{N};\mathbb{R... | Yes |
The zero section and the fibres of the cotangent bundle \( {T}^{ * }L \) with its canonical symplectic structure are Lagrangian because the canonical 1-form \( {\lambda }_{\text{can }} \) vanishes on these submanifolds (see Proposition 3.1.18). More generally, given a smooth submanifold \( Q \subset L \), the annihilat... | \[ T{Q}^{ \bot } = \left\{ {\left( {q,{v}^{ * }}\right) \in {T}^{ * }L{\left| q \in Q,{v}^{ * }\right| }_{{T}_{q}Q} = 0}\right\} \] is Lagrangian. | Yes |
Proposition 3.4.2 The graph \( {\Gamma }_{\sigma } \subset {T}^{ * }L \) of a 1-form \( \sigma \) on \( L \) is Lagrangian if and only if \( \sigma \) is closed. | Proof: The graph \( {\Gamma }_{\sigma } \) is the image of the embedding \( \sigma : L \rightarrow {T}^{ * }L \) . Hence the symplectic form \( - d{\lambda }_{\text{can }} \) vanishes on \( {\Gamma }_{\sigma } \) if and only if\n\n\[ 0 = {\sigma }^{ * }\left( {d{\lambda }_{\text{can }}}\right) = d\left( {{\sigma }^{ * ... | Yes |
Proposition 3.4.3 Let \( \\left( {M,\\omega }\\right) \) be a symplectic manifold and let \( \\psi : M \\rightarrow M \) be a diffeomorphism. Then \( \\psi \) is a symplectomorphism if and only if its graph\n\n\[ \n\\operatorname{graph}\\left( \\psi \\right) = \\{ \\left( {q,\\psi \\left( q\\right) }\\right) \\mid q \\... | Proof: Exercise. | No |
Example 3.4.7 (Chekanov torus) The set\n\n\\[ \n{L}_{\\mathrm{{Ch}}} \\mathrel{\\text{:=}} \\left\\{ {\\left. \\left( \\begin{matrix} \\left( {{e}^{s} + i{e}^{-s}t}\\right) \\cos \\left( \\theta \\right) \\\\ \\left( {{e}^{s} + i{e}^{-s}t}\\right) \\sin \\left( \\theta \\right) \\end{matrix}\\right) \\right| \\;\\begin... | To see this, denote by \\( \\theta \\in \\mathbb{R}/{2\\pi }\\mathbb{Z} \\) the coordinate on the unit circle \\( {S}^{1} \\subset \\mathbb{C} \\) and consider the diffeomorphism \\( {S}^{1} \\times \\mathbb{R} \\rightarrow {\\mathbb{R}}^{2} \\smallsetminus \\{ 0\\} : \\left( {\\theta, s}\\right) \\mapsto \\left( {{e}^... | No |
Theorem 3.4.10 (Symplectic neighbourhood theorem) For \( j = 0,1 \) let \( \left( {{M}_{j},{\omega }_{j}}\right) \) be a symplectic manifold with compact symplectic submanifold \( {Q}_{j} \) . Suppose that there is an isomorphism\n\n\[ \Phi : {\nu }_{{Q}_{0}} \rightarrow {\nu }_{{Q}_{1}} \]\n\nof the symplectic normal ... | Proof: First observe that \( \phi \) extends to a diffeomorphism \( {\phi }^{\prime } : \mathcal{N}\left( {Q}_{0}\right) \rightarrow \mathcal{N}\left( {Q}_{1}\right) \) such that \( d{\phi }^{\prime } \) induces the map \( \Phi \) on \( {\nu }_{{Q}_{0}} = T{Q}_{0}^{\omega } \) . Here we may take\n\n\[ {\phi }^{\prime }... | Yes |
Theorem 3.4.13 (Lagrangian neighbourhood theorem) Let \( \left( {M,\omega }\right) \) be a symplectic manifold and \( L \subset M \) be a compact Lagrangian submanifold. Then there exists a neighbourhood \( \mathcal{N}\left( {L}_{0}\right) \subset {T}^{ * }L \) of the zero section, a neighbourhood \( V \subset M \) of ... | Proof: The proof rests on the fact that the normal bundle of \( L \) in \( M \) is isomorphic to the tangent bundle. To define an explicit isomorphism, one may use a compatible complex structure \( J \) on the tangent bundle \( {TM} \), which exists by Proposition 2.6.4. (Such a complex structure is called an almost co... | Yes |
Proposition 3.4.14 Let \( \left( {M,\omega }\right) \) be a closed symplectic manifold. Then a neighbourhood of the identity in \( \operatorname{Symp}\left( M\right) \) can be identified with a neighbourhood of zero in the vector space of closed 1 -forms on \( M \) . | Proof: By Theorem 3.4.13, there exist a neighbourhood \( \mathcal{N}\left( \Delta \right) \subset M \times M \) of the diagonal \( \Delta \), a convex neighbourhood \( \mathcal{N}\left( {M}_{0}\right) \subset {T}^{ * }M \) of the zero section \( {M}_{0} \) , and a diffeomorphism \( \Psi : \mathcal{N}\left( \Delta \righ... | No |
Example 3.4.16 (Luttinger surgery) Let \( \left( {M,\omega }\right) \) be a symplectic 4-manifold without boundary and let \( \iota : {\mathbb{T}}^{2} \rightarrow M \) be a Lagrangian embedding of the standard 2-torus with coordinates \( x = \left( {{x}_{1},{x}_{2}}\right) \in {\mathbb{T}}^{2} = {\mathbb{R}}^{2}/{\math... | The cotangent bundle of the torus is \( {T}^{ * }{\mathbb{T}}^{2} = {\mathbb{T}}^{2} \times {\mathbb{R}}^{2} \) with coordinates \( \left( {{x}_{1},{x}_{2},{y}_{1},{y}_{2}}\right) \) and symplectic form \( {\omega }_{\text{can }} = d{x}_{1} \land d{y}_{1} + d{x}_{2} \land d{y}_{2} \) . The Lagrangian Neighbourhood Theo... | Yes |
Proposition 3.5.1 Let \( M \) be a manifold of dimension \( {2n} + 1 \) and \( \xi \subset {TM} \) be a transversally orientable hyperplane field.\n\n(i) Let \( \alpha \) be a 1 -form with \( \xi = \ker \alpha \) . Then \( {d\alpha } \) is nondegenerate on \( \xi \) if and only if\n\n\[ \alpha \land {\left( d\alpha \ri... | Proof: By Corollary 2.1.4, the 2-form \( {d\alpha } \) is nondegenerate on \( \xi = \ker \alpha \) if and only if \( {\left( d\alpha \right) }^{n} \) is nonzero on \( \xi \) . This is equivalent to \( \alpha \land {\left( d\alpha \right) }^{n} \neq 0 \) . If \( \alpha \) and \( {\alpha }^{\prime } \) are as in (ii), th... | Yes |
Let \( L \) be any compact manifold. Then the 1-jet bundle \[ {J}^{1}L \mathrel{\text{:=}} {T}^{ * }L \times \mathbb{R} \] is a contact manifold with the contact form \[ \alpha \mathrel{\text{:=}} {\lambda }_{\text{can }} - {dz} \] and the Reeb vector field \( Y = - \partial /\partial z \), where \( z \) is the real pa... | This is the analogue of the construction of Lagrangian submanifolds of \( {T}^{ * }L \) as the graphs of exact 1-forms. Any compact Legendrian submanifold \( L \) of any contact manifold \( M \) has a neighbourhood which is contactomorphic to a neighbourhood of the zero section in \( {J}^{1}L \) . | No |
A special case of Example 3.5.7 is the torus \( L = {\mathbb{T}}^{n} \). Its unit cotangent bundle \[ M \mathrel{\text{:=}} {\mathbb{T}}^{n} \times {S}^{n - 1} \] is a contact manifold with contact form \[ \alpha \mathrel{\text{:=}} \mathop{\sum }\limits_{{j = 1}}^{n}{y}_{j}d{x}_{j} \] where \( x = \left( {{x}_{1},\ldo... | Let \( {y}_{0} = \left( {{y}_{01},\ldots ,{y}_{0n}}\right) \in {S}^{n - 1} \) be any frequency vector with rationally dependent components, i.e. all the ratios \( {y}_{0i}/{y}_{0j} \) are rational. Then the manifold \[ L \mathrel{\text{:=}} \left\{ {\left( {x + j,{y}_{0}}\right) \mid \left\langle {x,{y}_{0}}\right\rang... | Yes |
Let \( \left( {V,\omega }\right) \) be a symplectic manifold and assume that the cohomology class of \( \omega \) admits an integral lift in \( {H}^{2}\left( {V;\mathbb{Z}}\right) \). Then there exists a circle bundle \( \pi : P \rightarrow V \) with Euler class \( - \left\lbrack \omega \right\rbrack \). | Denote the circle action on \( P \) by\n\n\[ P \times {S}^{1} \rightarrow P : \left( {p,{e}^{i\theta }}\right) \mapsto p{e}^{i\theta },\;\theta \in \mathbb{R}/{2\pi }\mathbb{Z}. \]\n\nThe action is generated by the vector field \( X \in \mathcal{X}\left( P\right) \), defined by\n\n\[ X\left( p\right) \mathrel{\text{:=}... | Yes |
The contact structure on the sphere in Example 3.5.9 is a special case of Example 3.5.11. It is obtained by considering the Hopf fibration | \[ {S}^{{2n} + 1} \rightarrow {\mathbb{{CP}}}^{n} \] (Compare with Example 4.3.3 below.) In that case the circle action is generated by the vector field \[ X \mathrel{\text{:=}} \mathop{\sum }\limits_{{j = 0}}^{n}\left( {{x}_{j}\frac{\partial }{\partial {y}_{j}} - {y}_{j}\frac{\partial }{\partial {x}_{j}}}\right) . \] ... | Yes |
Lemma 3.5.14 Let \( Y \) be the Reeb field of the contact form \( \alpha \). (i) The vector field \( X : M \rightarrow {TM} \) is a contact vector field if and only if there exists a function \( H : M \rightarrow \mathbb{R} \) such that \[ \iota \left( X\right) \alpha = H,\;\iota \left( X\right) {d\alpha } = \left( {\i... | Proof: If (3.5.7) holds then \( {\mathcal{L}}_{X}\alpha = {g\alpha } \) with \( g = \iota \left( Y\right) {dH} \). Conversely, assume \( {\mathcal{L}}_{X}\alpha = {g\alpha } \) and define \( H \mathrel{\text{:=}} \iota \left( X\right) \alpha \). Then \[ \iota \left( X\right) {d\alpha } = {\mathcal{L}}_{X}\alpha - {d\io... | Yes |
Proposition 3.5.23 (Symplectization) Let \( \left( {M,\xi }\right) \) be a transversally oriented contact manifold.\n\n(i) The set\n\n\[ \nW \mathrel{\text{:=}} \left\{ {\left( {q,{v}^{ * }}\right) \left| {\;\begin{array}{l} q \in M,{v}^{ * } \in {T}_{q}^{ * }M,\ker {v}^{ * } = {\xi }_{q},\text{ and } \\ v \in {T}_{q}M... | Proof: It follows directly from the definitions that \( W \) is a submanifold of \( {T}^{ * }M \) and that the map \( {\iota }_{\alpha } : \mathbb{R} \times M \rightarrow W \) is a diffeomorphism. It follows also from the definitions, with \( \dim M = {2n} + 1 \), that\n\n\[ \n{\omega }_{\alpha }^{n + 1} = {e}^{\left( ... | Yes |
The symplectization of the unit sphere \( M = {S}^{{2n} - 1} \) with the standard contact form \( \alpha = \frac{1}{2}\mathop{\sum }\limits_{{j = 1}}^{n}\left( {{y}_{j}d{x}_{j} - {x}_{j}d{y}_{j}}\right) \) (see Example 3.5.9) is symplectomorphic to \( {\mathbb{C}}^{n} \smallsetminus \{ 0\} \) with the standard symplect... | An explicit symplectomorphism is given by\n\n\[ \mathbb{R} \times {S}^{{2n} - 1} \rightarrow {\mathbb{C}}^{n} \smallsetminus \{ 0\} : \left( {s, x, y}\right) \mapsto {e}^{s/2}\left( {x + {iy}}\right) . \] | Yes |
Proposition 3.5.33 Let \( \\left( {W,\\omega }\\right) \) be a symplectic manifold of dimension \( {2n} \\geq 4 \) and let \( M \\subset W \) be a compact connected hypersurface of contact type. Then \( M \) has a preferred positive side into which any transverse Liouville vector field points. In particular, there is n... | Proof: The proof was explained to us by Janko Latschev. Let \( X \) and \( {X}^{\\prime } \) be Liouville vector fields near \( M \) that are transverse to \( M \) . Consider the contact forms\n\n\\[ \n{\\left. \\alpha \\mathrel{\\text{:=}} - \\iota \\left( X\\right) \\omega \\right| }_{M},\\;{\\alpha }^{\\prime } \\ma... | Yes |
Proposition 4.1.1 Let \( M \) be a smooth manifold.\n\n(i) For every nondegenerate 2-form \( \omega \) on \( M \) the spaces \( \mathcal{J}\left( {M,\omega }\right) \) and \( {\mathcal{J}}_{\tau }\left( {M,\omega }\right) \) are nonempty and contractible.\n\n(ii) Let \( J \) be an almost complex structure on \( M \) . ... | Proof: Proposition 4.1.1 follows immediately from Proposition 2.6.4. | No |
Every oriented hypersurface \( \sum \subset {\mathbb{R}}^{3} \) carries an almost complex structure \( J \) which it inherits from the vector product | Let \( \nu : \sum \rightarrow {S}^{2} \) be the Gauss map which associates to every point \( x \in \sum \) the outward unit normal vector\n\n\[ \nu \left( x\right) \bot {T}_{x}\sum \]\n\nThen the almost complex structure is given by the formula\n\n\[ {J}_{x}u \mathrel{\text{:=}} \nu \left( x\right) \times u. \]\n\nSuch... | Yes |
Lemma 4.1.14 Let \( M \) be a smooth manifold, let \( \omega \) be a nondegenerate 2 -form on \( M \), let \( J \in \mathcal{J}\left( {M,\omega }\right) \) be an \( \omega \) -compatible almost complex structure, denote by \( \langle \cdot , \cdot \rangle \mathrel{\text{:=}} \omega \left( {\cdot, J \cdot }\right) \) th... | Proof 1: Equation (4.1.9) follows by differentiating the identity \( {J}^{2} = - \mathbb{1} \) . To prove equation (4.1.10), choose vector fields \( X, Y, Z \) such that\n\n\[ X\left( q\right) = u,\;Y\left( q\right) = v,\;Z\left( q\right) = w, \]\n\nand differentiate the identity \( \langle {JY}, Z\rangle + \langle Y,{... | Yes |
Proposition 4.2.1 (i) \( {N}_{J} \) is a tensor, i.e. \( {N}_{J}\left( {{fX},{gY}}\right) = {fg}{N}_{J}\left( {X, Y}\right) \) for all \( X, Y \in \mathcal{X}\left( M\right) \) and all \( f, g \in {C}^{\infty }\left( M\right) \) . | Proof: According to the sign conventions of Remark 3.1.6 we can express the Lie bracket in local coordinates as \( \left\lbrack {X, Y}\right\rbrack = {\partial }_{Y}X - {\partial }_{X}Y \), where\n\n\[{\partial }_{X}Y = \mathop{\sum }\limits_{j}{\xi }_{j}\frac{\partial Y}{\partial {x}_{j}}\;\text{ when }\;X = \mathop{\... | Yes |
Theorem 4.2.2 (Integrability theorem) An almost complex structure \( J \) is integrable if and only if \( {N}_{J} = 0 \) . | Proof: See [371, Appendix 8] and [160, Chapter 2]. | No |
Lemma 4.2.4 The set of vector fields of type \( \left( {1,0}\right) \) on \( \left( {M, J}\right) \) is closed under the Lie bracket if and only if \( {N}_{J} = 0 \) . | Proof: If \( X \) and \( Y \) are real-valued vector fields on \( M \), then\n\n\[\n\left\lbrack {X - {iJX}, Y - {iJY}}\right\rbrack = \left\lbrack {X, Y}\right\rbrack - \left\lbrack {{JX},{JY}}\right\rbrack - i\left( {\left\lbrack {{JX}, Y}\right\rbrack + \left\lbrack {X,{JY}}\right\rbrack }\right) .\n\]\n\nThis vecto... | Yes |
Lemma 4.2.5 Let \( \omega \) be a nondegenerate 2-form on \( M \) and let \( J \in \mathcal{J}\left( {M,\omega }\right) \) . Let \( \nabla \) denote the Levi-Civita connection associated to the Riemannian metric \( {g}_{J} \mathrel{\text{:=}} \omega \left( {\cdot, J \cdot }\right) \) . Then the following are equivalent... | Proof: We prove that (i) implies (ii). If \( \nabla J = 0 \) then it follows from equation (4.1.11) in Lemma 4.1.14 that \( {d\omega } = 0 \) . Since \( \left\lbrack {X, Y}\right\rbrack = {\nabla }_{Y}X - {\nabla }_{X}Y \) and \( {\nabla }_{X}\left( {JY}\right) = \left( {{\nabla }_{X}J}\right) Y + J{\nabla }_{X}Y \) th... | Yes |
Theorem 4.2.6 Every almost complex structure on a two-dimensional manifold is integrable. | Proof: See page 165 and also [470, Appendix C.5] for another proof. | No |
Example 4.3.1 (Euclidean space) The Kähler manifold \( \left( {{\mathbb{R}}^{2n},{J}_{0},{\omega }_{0}}\right) \) can be identified with \( {\mathbb{C}}^{n} \) in such a way that \( {J}_{0} \) corresponds to multiplication by \( i \) . In complex geometry on \( {\mathbb{C}}^{n} \) it is convenient to deal with the comp... | \[ d{z}_{j} = d{x}_{j} + {id}{y}_{j},\;d{\bar{z}}_{j} = d{x}_{j} - {id}{y}_{j}. \] Note that these are complex-valued 1 -forms on \( {\mathbb{R}}^{2n} = {\mathbb{C}}^{n} \) . For complex-valued differential forms on \( {\mathbb{C}}^{n} \) the differential \( d : {\Omega }^{k} \rightarrow {\Omega }^{k + 1} \) can be con... | Yes |
Example 4.3.3 (Complex projective space) This is the space \( {\mathbb{{CP}}}^{n} \) of complex lines in \( {\mathbb-C}}^{n + 1} \) . Thus a point in \( {\mathbb{{CP}}}^{n} \) is the equivalence class of a nonzero complex \( \left( {n + 1}\right) \) -vector \( \left\lbrack z\right\rbrack = \left\lbrack {{z}_{0} : \cdot... | This is a real valued 2-form on \( {\mathbb-C}}^{n + 1} \smallsetminus \{ 0\} \) and it descends to a symplectic form on \( {\mathbb{{CP}}}^{n} \), which is denoted by \( {\omega }_{\mathrm{{FS}}} \) and called the Fubini-Study form. In other words, \( {\rho }_{\mathrm{{FS}}} = {\operatorname{pr}}^{ * }{\omega }_{\math... | Yes |
Example 4.3.6 Let \( \left( {V,\omega }\right) \) be a \( {2n} \) -dimensional symplectic vector space. Then the space \( \mathcal{J}\left( {V,\omega }\right) \) of \( \omega \) -compatible linear complex structures on \( V \) is a smooth manifold whose tangent space at \( J \in \mathcal{J}\left( {V,\omega }\right) \) ... | \[ {T}_{J}\mathcal{J}\left( {V,\omega }\right) = \left\{ {\widehat{J} \in \operatorname{End}\left( V\right) \mid \widehat{J}J + J\widehat{J} = 0,\omega \left( {\widehat{J}\cdot , \cdot }\right) + \omega \left( {\cdot ,\widehat{J} \cdot }\right) = 0}\right\} . \] | Yes |
The ruled surfaces form an interesting but fairly simple family of Kähler surfaces. These are complex surfaces \( X \) that fibre holomorphically over a Riemann surface \( \sum \) with fibre equal to \( {\mathbb{{CP}}}^{1} \) or, equivalently, \( {S}^{2} \). We now describe an easy way to construct families of ruled su... | \[ \mathbb{P}\left( {L \oplus \mathbb{C}}\right) \rightarrow \sum \] whose fibre at \( z \in \sum \) is the space \( \mathbb{P}\left( {{L}_{z} \oplus \mathbb{C}}\right) \cong {\mathbb{{CP}}}^{1} \) of all lines through the origin in \( {L}_{z} \oplus \mathbb{C} \) . | Yes |
Consider the hypersurface of degree \( d \) in \( {\mathbb{{CP}}}^{3} \)\n\n\[ \n{X}_{d} \mathrel{\text{:=}} \left\{ {\left\lbrack {{z}_{0} : \cdots : {z}_{3}}\right\rbrack \in {\mathbb{{CP}}}^{3} \mid {z}_{0}^{d} + {z}_{1}^{d} + {z}_{2}^{d} + {z}_{3}^{d} = 0}\right\} .\n\]\n\nThis is a smooth manifold. The Lefschetz h... | To see this, let \( L \rightarrow {\mathbb{{CP}}}^{3} \) be a holomorphic line bundle of degree \( d \) and choose a basis \( {s}_{0},\ldots ,{s}_{n} \) of the space of holomorphic sections of \( L \) . Then for every \( z \in {\mathbb{{CP}}}^{3} \) there exists a \( j \) such that \( {s}_{j}\left( z\right) \neq 0 \) .... | Yes |
A similar example in three complex dimensions is given by the following family of hypersurfaces in \( {\mathbb{{CP}}}^{4} \) :\n\n\[ \n{Z}_{d} \mathrel{\text{:=}} \left\{ {\left\lbrack {{z}_{0} : \cdots : {z}_{4}}\right\rbrack \in {\mathbb{{CP}}}^{4} \mid {z}_{0}^{d} + {z}_{1}^{d} + {z}_{2}^{d} + {z}_{3}^{d} + {z}_{4}^... | These manifolds are simply connected and have Betti numbers\n\n\[ \n{b}_{2} = {b}_{4} = 1,\;{b}_{3} = {d}^{4} - 5{d}^{3} + {10}{d}^{2} - {10d} + 4.\n\]\n\nThe identities \( {b}_{2} = {b}_{4} = 1 \) follow as before from the Lefschetz theorem on hyperplane sections. Hence \( {\pi }_{2}\left( {Z}_{d}\right) = \mathbb{Z} ... | Yes |
Example 4.4.5 (Adjunction formula) Let \( X \) be a Kähler surface (of 4 real dimensions) and \( \sum \subset X \) an embedded connected complex curve. Then, by Theorem 2.7.5, the first Chern class of its normal bundle \( {\nu }_{\sum } \) agrees with the self-intersection number of \( \sum \) . Since \( {T}_{\sum }X =... | \[ {2g}\left( \sum \right) - 2 = \sum \cdot \sum - \left\langle {{c}_{1}\left( {TX}\right) ,\left\lbrack \sum \right\rbrack }\right\rangle . \] | Yes |
Proposition 5.1.9 (i) Let \( I \subset \mathbb{R} \) be an interval and \( {\left\{ {\tau }_{\lambda }\right\} }_{\lambda \in I} \) be a family of symplectic forms on \( B \) such that\n\n\[ \left\lbrack {\tau }_{\lambda }\right\rbrack = \left\lbrack {\tau }_{\mu }\right\rbrack + \left( {\lambda - \mu }\right) {c}_{I} ... | Proof: Let \( {\phi }_{t}\left( p\right) = p \cdot {e}^{2\pi it} \) denote the principal circle action on \( P \) and let \( X : P \rightarrow {TP} \) denote the vector field which generates this action, namely\n\n\[ X\left( p\right) \mathrel{\text{:=}} p \cdot \left( {2\pi i}\right) \in {T}_{p}P. \]\n\nNow let \( {\ta... | Yes |
Lemma 5.1.10 Let \( \mathbb{R}/\mathbb{Z} \rightarrow \operatorname{Symp}\left( {M,\omega }\right) : t \mapsto {\psi }_{t} \) be a Hamiltonian circle action generated by a proper function \( H : M \rightarrow \mathbb{R} \) that is bounded below. Let \( {\lambda }_{0} \mathrel{\text{:=}} \) \( \min H \) . Then the level... | Proof: Because \( {\lambda }_{0} \) is the minimum of \( H \), the differential \( {dH} \) vanishes on the set \( Z = {H}^{-1}\left( {\lambda }_{0}\right) \) . Hence \( {X}_{H} = 0 \) on \( Z \) and all points of \( Z \) are fixed by \( {S}^{1} \) . The rest of the proof is an easy exercise which is left to the reader.... | No |
With this information, we can now build \( {S}^{1} \) -invariant symplectic forms on \( {S}^{2} \) -bundles \( \pi : M \rightarrow B \) which are associated to principal circle bundles \( P \rightarrow B \), provided that \( B \) supports a suitable family of symplectic forms. | Now suppose that \( {\tau }_{\lambda } \in {\Omega }^{2}\left( B\right) ,0 \leq \lambda \leq 1 \), is a family of symplectic forms on \( B \) that satisfies equation (5.1.2), where \( {c}_{I} \) is the first Chern class of the circle bundle \( P \rightarrow B \) . By slightly perturbing this family, we may assume in ad... | Yes |
Lemma 5.2.1 Let \( \mathrm{G} \rightarrow \operatorname{Symp}\left( {M,\omega }\right) : g \mapsto {\psi }_{g} \) be a symplectic action by a compact Lie group, let \( \mathfrak{g} \rightarrow \mathcal{X}\left( {M,\omega }\right) : \xi \mapsto {X}_{\xi } \) be the infinitesimal action, and let \( \mathfrak{g} \rightarr... | Proof: We assume first that \( \mathrm{G} \) is connected and prove that (ii) implies (i). Recall that \( {X}_{{g}^{-1}{\xi g}} = {\psi }_{g}^{ * }{X}_{\xi } \) and hence the functions \( {H}_{{g}^{-1}{\xi g}} \) and \( {H}_{\xi } \circ {\psi }_{g} \) generate the same Hamiltonian vector field. So their difference is c... | Yes |
Lemma 5.2.3 Assume that the action is weakly Hamiltonian and choose a linear map \( \mathfrak{g} \rightarrow {C}^{\infty }\left( M\right) : \xi \mapsto {H}_{\xi } \) such that \( {X}_{\xi } = {X}_{{H}_{\xi }} \) for all \( \xi \in \mathfrak{g} \) . Then there is a (unique) bilinear map \( \tau : \mathfrak{g} \times \ma... | Proof: The identity\n\n\[ {X}_{{H}_{\left\lbrack \xi ,\eta \right\rbrack }} = {X}_{\left\lbrack \xi ,\eta \right\rbrack } = \left\lbrack {{X}_{\xi },{X}_{\eta }}\right\rbrack = \left\lbrack {{X}_{{H}_{\xi }},{X}_{{H}_{\eta }}}\right\rbrack = {X}_{\left\{ {H}_{\xi },{H}_{\eta }\right\} } \]\n\nshows that the function \(... | Yes |
Lemma 5.2.5 Let \( \mathrm{G} \times M \rightarrow M : \left( {g, p}\right) \mapsto {\psi }_{g}\left( p\right) \) be a Hamiltonian group action, generated by an equivariant moment map \( \mu : M \rightarrow {\mathfrak{g}}^{ * } \), and let \( p \in M \) . Then the following holds.\n\n(i) The dual of the linear map \( {... | Proof: Equation (5.2.7) follows directly from (5.2.3). Moreover, by (5.2.3),\n\n\[ \n\operatorname{im}{L}_{p} \subset {\left( \ker d\mu \left( p\right) \right) }^{\omega } \n\]\n\nand, by (5.2.7),\n\n\[ \n\dim \operatorname{im}{L}_{p} = \dim \operatorname{im}{d\mu }\left( p\right) = \operatorname{codim}\ker {d\mu }\lef... | Yes |
Let \( \left( {V,\omega }\right) \) be a symplectic vector space and \( \mathrm{G} \mathrel{\text{:=}} \mathrm{{Sp}}\left( {V,\omega }\right) \) as in Example 5.3.5. Prove that the obvious action of \( \operatorname{Sp}\left( {V,\omega }\right) \) on \( V \) is generated by the moment map \( \mu : V \rightarrow \mathfr... | \[ \langle \mu \left( v\right), A\rangle = \frac{1}{2}\omega \left( {{Av}, v}\right) \] for \( v \in V \) and \( A \in \mathfrak{{sp}}\left( {V,\omega }\right) \) . | Yes |
Consider the action of a Lie group \( \mathrm{G} \) on its cotangent bundle \( {T}^{ * }\mathrm{G} \) which is induced by the right translations \( {R}_{{g}^{-1}} \) . In view of Exercise 3.1.21 the corresponding symplectomorphism \( {\psi }_{g} : {T}^{ * }\mathrm{G} \rightarrow {T}^{ * }\mathrm{G} \) is given by | \[ {\psi }_{g}\left( {h,{v}^{ * }}\right) = \left( {h{g}^{-1}, d{R}_{g}{\left( h{g}^{-1}\right) }^{ * }{v}^{ * }}\right) \] for \( {v}^{ * } \in {T}_{h}^{ * }\mathrm{G} \) . Now for every \( \xi \in \mathfrak{g} \) the 1-parameter group \( t \mapsto {R}_{\exp \left( {-{t\xi }}\right) } \) of diffeomorphisms of \( \math... | No |
Example 5.3.11 (Coadjoint orbits) Let G be a connected Lie group and let \( \mathcal{O} \subset {\mathfrak{g}}^{ * } \) be an orbit under the coadjoint action of \( \mathrm{G} \) . Then \( \mathcal{O} \) carries a natural symplectic structure. | To see this, note that the tangent space to \( \mathcal{O} \) at \( \eta \) is given by\n\n\[ \n{T}_{\eta }\mathcal{O} = \left\{ {\operatorname{ad}{\left( \xi \right) }^{ * }\eta \mid \xi \in \mathfrak{g}}\right\} \n\]\n\nwhere \( \operatorname{ad}\left( \xi \right) : \mathfrak{g} \rightarrow \mathfrak{g} \) denotes th... | Yes |
Example 5.3.18 (Flat connections over Riemann surfaces) This example is due to Atiyah and Bott [27]. Let \( \sum \) be a compact Riemann surface and \( \mathrm{G} \) be a compact Lie group with Lie algebra \( \mathfrak{g} = \operatorname{Lie}\left( \mathrm{G}\right) \) . Consider the space\n\n\[ \mathcal{A} \mathrel{\t... | This example generalizes naturally to nontrivial bundles as well as higher-dimensional Kähler manifolds (in place of Riemann surfaces). It also generalizes to product spaces \( \mathcal{A} \times {C}^{\infty }\left( {\sum, E}\right) \) consisting of pairs \( \left( {A, s}\right) \) , where \( A \) is a connection and \... | Yes |
Suppose that \( \left( {M,\omega }\right) \) is a closed symplectic \( {2n} \) -manifold. Then the space \( \mathcal{J}\left( {M,\omega }\right) \) of all almost complex structures on \( M \) that are compatible with \( \omega \) is itself an infinite-dimensional analogue of a Kähler manifold. Its tangent space at \( J... | The metric and symplectic form on \( {T}_{J}\mathcal{J}\left( {M,\omega }\right) \) are given by\n\n\[ \parallel \widehat{J}{\parallel }^{2} \mathrel{\text{:=}} \frac{1}{2}{\int }_{M}\operatorname{trace}\left( {\widehat{J}}^{2}\right) \frac{{\omega }^{n}}{n!},\;\Omega \left( {{\widehat{J}}_{1},{\widehat{J}}_{2}}\right)... | Yes |
For every coisotropic submanifold \( Q \subset M \) the distribution \( T{Q}^{\omega } \) is integrable. | Proof: Let \( X \) and \( Y \) be vector fields on \( Q \) with values in \( T{Q}^{\omega } \) and fix a point \( p \in Q \) . Given a tangent vector \( v \in {T}_{p}Q \) choose any vector field \( Z : Q \rightarrow {TQ} \) such that \( Z\left( p\right) = v \) . Since \( \omega \) is closed, we have\n\n\[ 0 = {d\omega ... | Yes |
Consider the unit sphere \( Q \mathrel{\text{:=}} {S}^{3} \) in \( \left( {{\mathbb{C}}^{2},{\omega }_{0}}\right) \). The isotropic leaves are the Hopf circles \( \left\{ {\left( {{e}^{it}{z}_{0},{e}^{it}{z}_{1}}\right) \mid t \in \mathbb{R}}\right\} \) and \( \bar{Q} = {\mathbb{{CP}}}^{1} \). For \( s \in \mathbb{R} \... | Moreover, if \( 0 < \left| s\right| < 1 \) the map \( \left( {{L}_{s} \smallsetminus \left\{ \left( {\sqrt{1 - {s}^{2}} + {is},0}\right) \right\} }\right) \cap Q \rightarrow {\mathbb{{CP}}}^{1} \) is an injective immersion, but not an embedding because it is not a homeomorphism to its image with the subspace topology. | Yes |
Consider the cotangent bundle of a closed manifold \( Y \). In [402], Laudenbach and Sikorav show that given any Lagrangian submanifold \( L \subset {T}^{ * }Y \) that is Hamiltonian isotopic to the zero section there exists a fibre bundle \( E \rightarrow Y \) and a smooth function \( S : E \rightarrow \mathbb{R} \) s... | This example is discussed in detail in Section 9.4; see in particular Proposition 9.4.4. | No |
Proposition 5.4.13 Assume (RG). Then zero is a regular value of the moment map, its preimage \( {\mu }^{-1}\left( 0\right) \) is a regular coisotropic submanifold of \( M \), the isotropic leaves are the orbits of \( \mathrm{G} \), and the quotient space\n\n\[ M//\mathrm{G} \mathrel{\text{:=}} {\mu }^{-1}\left( 0\right... | Proof: The proof has three steps.\n\nStep 1. If zero is a regular value of \( \mu \), then \( {\mu }^{-1}\left( 0\right) \) is a coisotropic submanifold of \( M \) and the isotropic leaves are the G-orbits in \( {\mu }^{-1}\left( 0\right) \).\n\nDenote the G-orbit of \( p \in M \) by \( \mathcal{O}\left( p\right) \math... | Yes |
Proposition 5.4.15 Assume G acts freely and properly on \( {\mu }^{-1}\left( \mathcal{O}\right) \) . Then \( \mu \) is transverse to \( \mathcal{O},{\mu }^{-1}\left( \mathcal{O}\right) \) is a smooth submanifold of \( M \), and the quotient space\n\n\[ \n{M}_{\mathcal{O}} \mathrel{\text{:=}} {\mu }^{-1}\left( \mathcal{... | Proof: Consider the diagonal action of \( \mathrm{G} \) on the product symplectic manifold \( \left( {\widetilde{M},\widetilde{\omega }}\right) \) defined by\n\n\[ \n\widetilde{M} \mathrel{\text{:=}} M \times \mathcal{O},\;\widetilde{\omega} \mathrel{\text{:=}} \omega \oplus \left( {-{\omega }_{\mathcal{O}}}\right) .\n... | Yes |
Theorem 5.5.1 (Atiyah-Guillemin-Sternberg) Let \( \left( {M,\omega }\right) \) be a closed connected symplectic manifold and let \( {\mathbb{T}}^{m} \rightarrow \operatorname{Symp}\left( {M,\omega }\right) : \theta \mapsto {\psi }_{\theta } \) be a Hamiltonian torus action generated by a moment map \( \mu : M \rightarr... | Proof: See page 237. | No |
Consider the action of the torus \( {\mathbb{T}}^{n} \) on \( {\mathbb{{CP}}}^{n} \) given by\n\n\[ \left( {{\theta }_{1},\ldots ,{\theta }_{n}}\right) \cdot \left\lbrack {{z}_{0} : {z}_{1} : \cdots : {z}_{n}}\right\rbrack = \left\lbrack {{z}_{0} : {e}^{-{2\pi i}{\theta }_{1}}{z}_{1} : \cdots : {e}^{-{2\pi i}{\theta }_... | By Exercise 5.3.16, the moment map \( \mu : {\mathbb{{CP}}}^{n} \rightarrow {\mathbb{R}}^{n} \) has the form\n\n\[ \mu \left( \left\lbrack {{z}_{0} : {z}_{1} : \cdots : {z}_{n}}\right\rbrack \right) = \pi \left( {\frac{{\left| {z}_{1}\right| }^{2}}{\parallel z{\parallel }^{2}},\ldots ,\frac{{\left| {z}_{n}\right| }^{2}... | Yes |
Schur [566] proved that if \( {\lambda }_{1},\ldots ,{\lambda }_{n} \) are the eigenvalues of a Hermitian matrix \( A = {A}^{ * } \in {\mathbb{C}}^{n \times n} \), then the vector \( a = \left( {{a}_{1},\ldots ,{a}_{n}}\right) \in {\mathbb{R}}^{n} \) of diagonal entries of \( A \) lies in the convex hull of the points\... | Conversely, it was proved by Horn [332] that for any vector \( a \) in this convex hull there exists a Hermitian matrix \( A \) with diagonal entries \( {a}_{j} \) and eigenvalues \( {\lambda }_{j} \) . As noted by Atiyah [26], these results can be obtained as a special case of Theorem 5.5.1.\n\nLet \( \mathrm{G} = \ma... | Yes |
Lemma 5.5.6 There exists an almost complex structure \( J \) on \( M \) that is compatible with \( \omega \) and invariant under the action of the torus in the sense that\n\n\[{\psi }_{\theta }^{ * }J = J\]\n\nfor every \( \theta \in {\mathbb{T}}^{m} \) . | Proof: Fix a Riemannian metric \( {g}_{0} \) on \( M \) and average the metrics \( {\psi }_{\theta }^{ * }{g}_{0} \) over the torus to obtain an invariant metric \( g \) . The image of this metric under the map \( \mathfrak{{Met}}\left( V\right) \rightarrow \mathcal{J}\left( {V,\omega }\right) : g \mapsto {J}_{g,\omega... | Yes |
Lemma 5.5.7 Let \( \mathrm{G} \subset {\mathbb{T}}^{m} \) be a subgroup. Then the fixed point set\n\n\[ \n\operatorname{Fix}\left( \mathrm{G}\right) = \mathop{\bigcap }\limits_{{\theta \in \mathrm{G}}}\operatorname{Fix}\left( {\psi }_{\theta }\right)\n\]\n\nis a symplectic submanifold of \( M \) . | Proof: Let \( x \in \operatorname{Fix}\left( \mathrm{G}\right) \) and, for \( \theta \in \mathrm{G} \), denote the differential of the symplec-tomorphism \( {\psi }_{\theta } \) at \( x \) by\n\n\[ \n{\Psi }_{\theta } \mathrel{\text{:=}} d{\psi }_{\theta }\left( x\right) : {T}_{x}M \rightarrow {T}_{x}M.\n\]\n\nThese ma... | Yes |
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