Thesis (Ph. D.)--University of Washington, 2003. / Vita. Includes bibliographical references (p. 174-178).

The thesis deals with contact manifolds and their groups of transformations and relatedly with contact fibre bundles. We apply the framework of convenient calculus on in finite dimensional smooth manifolds to study the Chern-Weil theory of groups of strict contactomorphisms producing several non-vanishing type results on the cohomology of the classifying spaces of these groups. Moreover, we prove that the space of isocontact embeddings of one contact manifold to another can be given the structure of a smooth manifold and a principal bundle. Using this we describe a particular smooth model of the classifying space for the group Cont+(M; ) of (co-orientation preserving) contactomorphisms of a closed contact manifold (M; ). Lastly, we show that the standard action of the unitary group U(2) on the standard contact 3-sphere S3 induces a homotopy equivalence Cont+(S3; std) ' U(2).

Low, Ho Chi. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2013. / Includes bibliographical references (leaves 76-79). / Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web. / Abstracts also in Chinese.

Ku Wah Kwan. / Thesis (M.Phil.)--Chinese University of Hong Kong, 2004. / Includes bibliographical references (leaves 61-64). / Abstracts in English and Chinese. / Chapter 1 --- Basic 3-Dimensional Contact Geometry --- p.5 / Chapter 1.1 --- Introduction --- p.5 / Chapter 1.2 --- Contact Structure --- p.7 / Chapter 1.3 --- Darboux's Theorem --- p.11 / Chapter 1.4 --- Characteristic Foliation --- p.13 / Chapter 1.5 --- More About S3 with The Standard Contact Structure --- p.16 / Chapter 2 --- Legendrian Knots --- p.18 / Chapter 2.1 --- Basic Definition --- p.18 / Chapter 2.2 --- Front Projection --- p.19 / Chapter 2.3 --- Classical Legendrian Knot Invariants --- p.22 / Chapter 2.3.1 --- Thurston-Bennequin Invariant --- p.22 / Chapter 2.3.2 --- Rotation Number --- p.23 / Chapter 2.4 --- Stabilization --- p.24 / Chapter 3 --- Convex Surface Theory --- p.26 / Chapter 3.1 --- Contact Vector Field --- p.26 / Chapter 3.2 --- Convex Surfaces --- p.29 / Chapter 3.3 --- Flexibility of Characteristic Foliation --- p.34 / Chapter 3.4 --- Bennequin Inequality --- p.36 / Chapter 3.5 --- Bypass --- p.38 / Chapter 3.5.1 --- Modification of Dividing Curves through Bypass --- p.39 / Chapter 3.5.2 --- Relation of Bypass and Stabilizing Disk --- p.40 / Chapter 3.5.3 --- Finding Bypass --- p.40 / Chapter 3.6 --- Tight Contact Structures on Solid Tori --- p.41 / Chapter 4 --- Classification Results --- p.42 / Chapter 4.1 --- Unknot --- p.43 / Chapter 4.2 --- Positive Torus Knot --- p.45 / Chapter 5 --- Transverse Knots --- p.50 / Chapter 5.1 --- Basic Definition --- p.50 / Chapter 5.2 --- Self-linking Number --- p.54 / Chapter 5.3 --- Relation between Transverse Knot and Legendrian Knot --- p.55 / Chapter 5.4 --- Classification of Unknot and Torus Knot --- p.57 / Chapter 6 --- Recent Development --- p.60 / Bibliography --- p.61

Thesis (Ph. D.)--Michigan State University. Dept. of Mathematics, 2008. / Title from PDF t.p. (viewed on July 23, 2009) Includes bibliographical references (p. 106-108). Also issued in print.

In this thesis, we study Legendrian and transverse isotopy problem for cabled knot types. We give two structural theorems to describe when the (r,s)- cable of a Legendrian simple knot type K is also Legendrian simple. We then study the same problem for cables of the positive trefoil knot. We give a complete classification of Legendrian and transverse cables of the positive trefoil. Our results exhibit many new phenomena in the structural understanding of Legendrian and transverse knots. we then extend these results to the other positive torus knots. The key ingredient in these results is to find necessary and sufficient conditions on maximally thickened contact neighborhoods of the positive torus knots in three sphere.

This thesis consists of a summary of two papers dealing with questions related to Legendrian submanifolds of contact manifolds together with exact Lagrangian cobordisms between Legendrian submanifolds. The focus is on studying Legendrian submanifolds from the perspective of their handle decompositions. The techniques used are mainly from Symplectic Field Theory. In Paper I, a series of examples of Legendrian surfaces in standard contact 5-space are studied. For every g > 0, we produce g+1 Legendrian surfaces of genus g, all with g+1 transverse Reeb chords, which lie in distinct Legendrian isotopy classes. For each g, exactly one of the constructed surfaces has a Legendrian contact homology algebra admitting an augmentation. Moreover, it is shown that the same surface is the only one admitting a generating family. Legendrian contact homology with Novikov coefficients is used to classify the different Legendrian surfaces. In particular, we study their augmentation varieties. In Paper II, the effect of a Legendrian ambient surgery on a Legendrian submanifold is studied. Given a Legendrian submanifold together which certain extra data, a Legendrian ambient surgery produces a Legendrian embedding of the manifold obtained by surgery on the original submanifold. The construction also provides an exact Lagrangian handle-attachment cobordism between the two submanifolds. The Legendrian contact homology of the submanifold produced by the Legendrian ambient surgery is then computed in terms of pseudo-holomorphic disks determined by data on the original submanifold. Also, the cobordism map induced by the exact Lagrangian handle attachment is computed. As a consequence, it is shown that a sub-critical standard Lagrangian handle attachment cobordism induces a one-to-one correspondence between the augmentations of the Legendrian contact homology algebras of its two ends.

We will discuss what is known about the construction of contact structures via branched covers, emphasizing the search for universal transverse knots. Recall that a topological knot is called universal if all 3-manifold can be obtained as a cover of the 3-sphere branched over that knot. Analogously one can ask if there is a transverse knot in the standard contact structure on S³ from which all contact 3-manifold can be obtained as a branched cover over this transverse knot. It is not known if such a transverse knot exists.

Στη μεταπτυχιακή αυτή διπλωματική εργασία, αρχικά εισάγουμε τις έννοιες των μετρικών πολλαπλοτήτων σχεδόν επαφής και των μετρικών πολλαπλοτήτων επαφής, δίνοντας και μερικά παραδείγματα από κάθε κατηγορία. Στη συνέχεια, αναφέρουμε και αποδεικνύουμε λεπτομερώς μερικές βασικές γεωμετρικές ιδιότητες που χαρακτηρίζουν τις μετρικές πολλαπλότητες επαφής και, οι οποίες, εμπλέκουν τον τανυστή καμπυλό- τητας. Τέλος, δίνεται έμφαση σε ειδικές κατηγορίες μετρικών πολλαπλοτήτων επαφής που παρουσιάζουν ιδιαίτερο γεωμετρικό ενδιαφέρον και, κυρίως, είναι: πολλαπλότητες K- επαφής, πολλαπλότητες του Sasaki, (κ, μ) – πολλαπλότητες επαφής και μετρικές πολλαπλότητες Η – επαφής. / In this Master Thesis, we initially introduce the notions of almost contact metric manifolds and contact metric manifolds, giving some examples from each category. In sequel, we mention and prove explicitly some basic geometric properties of contact metric manifolds, which involve the curvature tensor. Summarizing, we focus on special classes of contact metric manifolds which have particular geometric interest and, mainly, are: K – contact manifolds, Sasakian manifolds, (κ, μ) – contact manifolds and H – contact metric manifolds.

This thesis consists of three scientific papers dealing with invariants of Legendrian and Lagrangian submanifolds. Besides the scientific papers, the thesis contains an introduction to contact and symplectic geometry, and a brief outline of Symplectic field theory with focus on Legendrian contact homology. In Paper I we give an orientation scheme for moduli spaces of rigid flow trees in Legendrian contact homology. The flow trees can be seen as the adiabatic limit of sequences of punctured pseudo-holomorphic disks with boundary on the Lagrangian projection of the Legendrian. So to equip the trees with orientations corresponds to orienting the determinant line bundle of the dbar-operator over the space of Lagrangian boundary conditions on the punctured disk. We define an  orientation of this line bundle and prove that it is well-defined in the limit. We also prove that the chosen orientation scheme gives rise to a combinatorial algorithm for computing the orientation of the trees, and we give an explicit description of this algorithm. In Paper II we study exact Lagrangian cobordisms with cylindrical Legendrian ends, induced by Legendrian isotopies. We prove that the combinatorially defined DGA-morphisms used to prove invariance of Legendrian contact homology for Legendrian knots over the integers can be derived analytically.  This is proved using the orientation scheme from Paper I together with a count of abstractly perturbed flow trees  of the Lagrangian cobordisms. In Paper III we prove a flexibility result for closed, immersed Lagrangian submanifolds in the standard symplectic plane.