Künneth geometry : symplectic manifolds and their Lagrangian foliations / M. J. D. Hamilton, D. Kotschick.

Format:

Book

Author/Creator:

Hamilton, Mark J. D., author.

Kotschick, D. (Dieter), 1963- author.

Series:

London Mathematical Society lecture note series ; 489.

London Mathematical Society lecture note series ; 489

Language:

English

Subjects (All):

Symplectic geometry.

Symplectic manifolds.

Foliations (Mathematics).

Physical Description:

1 online resource (viii, 191 pages) : digital, PDF file(s).

Edition:

1st ed.

Place of Publication:

Cambridge ; New York, NY : Cambridge University Press, 2024.

Summary:

This clear and elegant text introduces Künneth, or bi-Lagrangian, geometry from the foundations up, beginning with a rapid introduction to symplectic geometry at a level suitable for undergraduate students. Unlike other books on this topic, it includes a systematic development of the foundations of Lagrangian foliations. The latter half of the text discusses Künneth geometry from the point of view of basic differential topology, featuring both new expositions of standard material and new material that has not previously appeared in book form. This subject, which has many interesting uses and applications in physics, is developed ab initio, without assuming any previous knowledge of pseudo-Riemannian or para-complex geometry. This book will serve both as a reference work for researchers, and as an invitation for graduate students to explore this field, with open problems included as inspiration for future research.

Contents:

1. Introduction page. 1.1. Motivation ; 1.2. What is in this book? ; 1.3. How to read this book ; 1.4. What is not in this book

2. Linear algebra and bundle theory. 2.1. Linear algebra ; 2.2. Symplectic and complex vector bundles ; 2.3. Künneth vector bundles

3. Symplectic geometry. 3.1. Symplectic forms on manifolds ; 3.2. The Moser method ; 3.3. Neighbourhoods of Lagrangian submanifolds

4. Foliations and connections. 4.1. Foliations and flat bundles ; 4.2. Lagrangian foliations ; 4.3. Integrability and torsion ; 4.4. Almost product structures

5. Künneth structures. 5.1. Basic definitions and results ; 5.2. The induced metric and the automorphism group ; 5.3. Examples ; 5.4. Remarks on terminology

6. The Künneth connection. 6.1. The canonical connection ; 6.2. The torsion of the Künneth connection ; 6.3. Para-Kahler structures

7. The curvature of a Künneth structure. 7.1. Symmetries of the Künneth curvature tensor ; 7.2. Concrete curvature computations ; 7.3. The Ricci and scalar curvatures ; 7.4. Künneth–Einstein structures ; 7.5. Kähler–Künneth structures

8. Hypersymplectic geometry. 8.1. Simple recursion operators ; 8.2. Hypersymplectic structures ; 8.3. An S¹-family of Künneth structures ; 8.4. The Ricci curvature of hypersymplectic structures

9. Nilmanifolds. 9.1. Nilpotent and solvable lie algebras ; 9.2. Nil- and infra-nilmanifolds ; 9.3. Lagrangian foliations and Künneth structures ; 9.4. Low-dimensional examples ; 9.5. Hypersymplectic structures on nilmanifolds ; 9.6. Anosov symplectomorphisms ; 9.7. Solvmanifolds

10. Four-manifolds. 10.1. Classical invariants ; 10.2. Almost Künneth structures ; 10.3. The integrable case ; 10.4. Constraints on symplectic Calabi–Yau four-manifolds ; 10.5. Künneth–Einstein structures ; 10.6 T²-bundles over T² ; 10.7 Open problems.

Notes:

Title from publisher's bibliographic system (viewed on 15 Dec 2023).

Includes bibliographical references and index.

ISBN:

1-108-90561-7

1-108-90297-9