By Bloch S. (ed.)
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Additional info for Algebraic Geometry - Bowdoin 1985, Part 1
The title of our favorite Poincar´e paper  became there “A certain theorem of geometry”. 34 Joined by a one-parameter family of symplectomorphisms with single valued (but timedependent) Hamiltonians. I. Arnold. I. Arnold and Symplectic Topology 21 And he listed the results obtained so far—a state of the art in 1986. That is, the torus ([37, 38, 40]), the surfaces ([43, 45]), the complex projective space (), (many) K¨ahler manifolds of negative curvature ([45, 60]), diﬀeomorphisms that are C 0 -close to the identity ().
The contact condition for ξ (with respect to the product orientation) is equivalent to ( ) ut dβt + βt ∧ (dut − β˙ t ) > 0 t where β˙ t denotes ∂β ∂t . This condition is a non-linear partial diﬀerential relation which is not so simple. The main thrust of the following discussion will be to simplify it by ﬁxing some of the terms. 3. Reconstruction Lemmas The easiest case is to ﬁx the whole family βt . In this case the contact condition ( ) is only about the family ut and becomes convex. In particular the space of solutions ut is connected and we get: Lemma 12 (Global reconstruction).
Of course they can all be used as the local model in the deﬁnition of a contact structure. 36 P. Massot The Canonical Contact Structure on S3 . We have already met the canonical contact structure on S3 coming from the canonical contact structures on ST ∗ S2 . One can prove that it is also • the orthogonal of the Hopf circles for the round metric, • a left-invariant contact structure on the Lie group SU(2), • T S3 ∩ JT S3 when S3 is seen as the boundary of the unit ball in C2 and J denotes the action of multiplication by i in T C2 .
Algebraic Geometry - Bowdoin 1985, Part 1 by Bloch S. (ed.)