Algebraic Geometry. Proc. conf. Ann Arbor, 1981 by I. Dolgachev

By I. Dolgachev

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Check that ϕ∗ ω = θ∗ ψ ∗ ω = ω0 . ω1 Remark. 14 classifies lagrangian embeddings: up to symplectomorphism, the set of lagrangian embeddings is the set of embeddings of manifolds into their cotangent bundles as zero sections. The classification of isotropic embeddings was also carried out by Weinstein in [45, 46]. An isotropic embedding of a manifold X into a symplectic manifold (M, ω) is a closed embedding i : X → M such that i∗ ω = 0. Weinstein showed that neighbourhood equivalence of isotropic embeddings is in one-to-one correspondence with isomorphism classes of symplectic vector bundles.

A riemannian manifold (X, g) is geodesically convex if every point x is joined to every other point y by a unique (up to reparametrization) minimizing geodesic. 1 In riemannian geometry, a geodesic is a curve which locally minimizes distance and whose velocity is constant. 36 LECTURE 3. GENERATING FUNCTIONS Example. On X = Rn with T X Rn × Rn , let gx (v, w) = v, w , gx (v, v) = 2 |v| , where ·, · is the euclidean inner product, and | · | is the euclidean norm. Then (Rn , ·, · ) is a geodesically convex riemannian manifold, and the riemannian distance is the usual euclidean distance d(x, y) = |x − y|.

56 LECTURE 4. HAMILTONIAN FIELDS Modulo 2π in θ, the function H has exactly two critical points: a critical point s where H vanishes, and a critical point u where H equals c. These points are called the stable and unstable points of H, respectively. This terminology is justified by the fact that a trajectory of the hamiltonian vector field of H whose initial point is close to s stays close to s forever, whereas this is not the case for u. ) The spherical pendulum is a mechanical system consisting of a massless rigid rod of length , fixed at one end, whereas the other end has a plumb bob of mass m, which may oscillate freely in all directions.

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