An introduction to contact topology by Hansjörg Geiges

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.

Show description

Read or Download An introduction to contact topology PDF

Best topology books

Solitons: Differential equations, symmetries, and infinite-dimensional algebras

This booklet investigates the excessive measure of symmetry that lies hidden in integrable platforms. for this reason, differential equations bobbing up from classical mechanics, equivalent to the KdV equation and the KP equations, are used right here through the authors to introduce the inspiration of an unlimited dimensional transformation team performing on areas of integrable structures.

Rotations, Quaternions, and Double Groups (Oxford Science Publications)

This exact monograph treats finite aspect teams as subgroups of the entire rotation crew, supplying geometrical and topological tools which permit a distinct definition of the quaternion parameters for all operations. an enormous characteristic is an common yet entire dialogue of projective representations and their software to the spinor representations, which yield nice merits in precision and accuracy over the extra classical double workforce process.

Additional resources for An introduction to contact topology

Sample text

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 [204]. ) 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).

Download PDF sample

Rated 4.20 of 5 – based on 17 votes