By Hansjörg Geiges
This article on touch topology is the 1st accomplished advent to the topic, together with fresh awesome purposes in geometric and differential topology: Eliashberg's evidence of Cerf's theorem through the class of tight touch constructions at the 3-sphere, and the Kronheimer-Mrowka facts of estate P for knots through symplectic fillings of touch 3-manifolds. beginning with the fundamental differential topology of touch manifolds, all facets of three-d touch manifolds are handled during this e-book. One outstanding characteristic is an in depth exposition of Eliashberg's category of overtwisted touch buildings. Later chapters additionally take care of higher-dimensional touch topology. the following the focal point is on touch surgical procedure, yet different buildings of touch manifolds are defined, resembling open books or fibre hooked up sums. This booklet serves either as a self-contained creation to the topic for complex graduate scholars and as a reference for researchers.
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Additional resources for An introduction to contact topology
The following proposition gives a more concrete geometric interpretation of this statement. 12 Let (M, ξ) be a contact manifold of dimension 2n + 1 and L ⊂ (M, ξ) an isotropic submanifold. Then dim L ≤ n. 5 The geodesic ﬂow and Huygens’ principle 35 Proof Write i for the inclusion of L in M and let α be an (at least locally deﬁned) contact form deﬁning ξ. Then the condition for L to be isotropic becomes i∗ α ≡ 0. It follows that i∗ dα ≡ 0. In particular, Tp L ⊂ ξp is an isotropic subspace of the 2n–dimensional symplectic vector space (ξp , dα|ξ p ).
Consider the commutative diagram T T ∗ B −−−−→ T ∗ B π Tπ TB −−−−→ B and deﬁne a diﬀerential 1–form λ on T ∗ B by λu = u ◦ T π for u ∈ T ∗ B. This 1–form is called the Liouville form on T ∗ B. 1 In local coordinates q = (q1 , . . , qn ) on the manifold B and P1: xxx CUUK1064-McKenzie 20 December 7, 2007 20:6 Facets of contact geometry dual coordinates p = (p1 , . . , pn ) on the ﬁbres of T ∗ B, the Liouville form λ is equal to n pj dqj =: p dq. e. section of the bundle T ∗ B → B) one has τ = τ ∗ λ.
For a comprehensive treatment of geodesic ﬂows P1: xxx CUUK1064-McKenzie December 7, 2007 20:6 Facets of contact geometry 26 in the context of the theory of dynamical systems see the monograph by Paternain . ) We also consider the dual ﬂow on the cotangent bundle; here the discussion ties up with the space of contact elements and yields a simple contact geometric proof of Huygens’ principle concerning the propagation of wave fronts. 1 Let B be a manifold with a Riemannian metric g. There is a unique vector ﬁeld G on the tangent bundle T B whose trajectories are of the form t → γ(t) ˙ ∈ Tγ (t) B ⊂ T B, where γ is a geodesic on B (not necessarily of unit speed).