Existence of Almost Contact Structures on Manifolds with G2-structures and Generalizations

54 Pages·2012·0.29 MB·English

Existence of Almost Contact Structures on
Manifolds with G -structures and Generalizations
2
by
Hyunjoo Cho
Submitted in Partial Fulfillment
of the
Requirements for the Degree
Doctor of Philosophy
Supervised by
Professor Sema Salur
Department of Mathematics
Arts, Sciences and Engineering
School of Arts and Sciences
University of Rochester
Rochester, New York
2012
ii
Curriculum Vitae
Hyunjoo Cho was born in Gochang, Geonbuk, South Korea. She earned a Bach-
elor’s degree in Mathematics Education from Kongju National University, South
Korea and Master of Science degrees in Mathematics from Ewha Womans Uni-
versity, South Korea and Michigan State University. She joined the Department
of Mathematics at the University of Rochester in September 2007 and studied
differential geometry and calibrated geometry under the direction of Associate
Professor Sema Salur.
iii
Acknowledgments
I am deeply honored to thank my advisor, Associate Professor Sema Salur, for
showing me her great enthusiasm for mathematics and encouraging me tremen-
dously not only as my advisor but as my colleague. I also thank Visiting Assistant
ProfessorFiratArikanforteachingmecontactgeometrywhileweworkedtogether
and I appreciate Professor Selman Akbulut for helping me continue my academic
work. In addition, I would like to thank Professor Michael Gage teaching me
differential geometry and the Department of Mathematics at the University of
Rochester for funding me. Lastly, I would like to thank my family because I
would not be here without their unwavering support.
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Abstract
This dissertation consists of two main results. First, we investigate the relation-
ship between almost contact structures and G -structures on seven-dimensional
2
Riemannian manifolds: we show that any manifold with a G -structure admits
2
an almost contact structure by proving that there exists an almost contact struc-
ture on any seven-dimensional manifold with a spin structure. We also construct
explicit almost contact structures on manifolds with G -structures. Moreover,
2
we extend any almost contact structure on an associative submanifold to a G -
2
manifold. The second part of this thesis shows that contact structures and G -
2
structures are compatible in certain ways if contact structures exist on a manifold
with G -structures. We also introduce a new structure, which we call a contact-
2
G -structure, on a seven-dimensional manifold. Finally, we present examples of a
2
manifold with (torsion free) G -structures with compatible contact structures.
2
v
Table of Contents
Curriculum Vitae ii
Acknowledgments iii
Abstract iv
1 Introduction 1
2 Almost contact structures and Contact structures 4
2.1 Almost contact structures . . . . . . . . . . . . . . . . . . . . . . 4
2.2 Contact structures . . . . . . . . . . . . . . . . . . . . . . . . . . 10
3 G -structures 16
2
4 Existence of almost contact structures on manifolds with G -
2
structures 22
4.1 Almost contact structures on 7-manifolds with spin structures . . 22
4.2 Explicit construction of almost contact metric structures on mani-
folds with G -structures . . . . . . . . . . . . . . . . . . . . . . . 24
2
4.3 Extension of almost contact structures on G -manifolds . . . . . . 26
2
vi
5 Compatibility 29
5.1 Compatible contact structures with the G -structure . . . . . . . 29
2
5.2 Contact-G -structures on 7-manifolds . . . . . . . . . . . . . . . . 34
2
6 Examples 41
6.1 CY ×S1 or CY ×R . . . . . . . . . . . . . . . . . . . . . . . . . 41
6.2 W ×S1 or W ×R . . . . . . . . . . . . . . . . . . . . . . . . . . 42
6.3 R3 ×K4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
6.4 T∗M3 ×R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
Bibliography 46
1
1 Introduction
A seven-dimensional Riemannian manifold is said to admit a G -structure if the
2
structure group of its frame bundle reduces to the group G . This implies that we
2
can smoothly attach each tangent space of a manifold with a G -structure at any
2
point p to the imaginary octonions im(O) ∼= R7. In particular, G -manifolds are
2
manifolds with torsion free G -structures and are equivalent to manifolds with
2
holonomy groups in G . In 1966, E. Bonan constructed the manifold with a
2
parallel 3-form and showed that this manifold was Ricci-flat [1]. In 1989, the
first examples of non-compact seven-dimensional manifolds with holonomy group
G were constructed by R. Bryant and S. Salamon [4] and after that, the first
2
compact seven-dimensional manifolds with holonomy group G were introduced
2
byD.Joyce[13]. Althoughtherearevariouswaystoinvestigateseven-dimensional
manifolds with (torsion free) G -structures, we wish to provide elegant results to
2
describe the relations between the contact structures and G -structures on seven-
2
dimensional manifolds. In this work, we focus on seven-dimensional manifolds not
only equipped with contact structures which are compatible with G -structures,
2
but also with almost contact structures. This will be a novel way to study G -
2
geometry as well as to study high dimensional contact geometry.
Before we present the relations between contact structures and G -structures,
2
we first observe manifolds with G -structures in the context of almost contact
2
2
metricmanifolds. OurinvestigationofrelationsbetweenG -structuresandalmost
2
contactstructuresisbasedonTheorem1: ifamanifoldadmitsacontactstructure,
then there exists an almost contact structure satisfying a certain metric condition.
We begin with a brief survey of almost contact structures in section 2.1. In 1959,
J. Gray introduced the notion of almost contact manifolds by odd-dimensional
manifolds whose structure group of tangent bundle can be reduced to U(n) × 1
[10]. Later, in1960, S.Sasakiintroducedanequivalentdefinitionofalmostcontact
manifolds[19]. Proposition3showsthatthesetwodefinitionsofanalmostcontact
structure are equivalent. In section 2.2, we review a (global) contact distribution
and present an example of a contact manifold. Contact geometry was revealed
in 1896 in the work of Sophus Lie and it has been used as an important tool to
study odd-dimensional manifolds as an analogue of symplectic geometry in even-
dimensionalmanifolds. See[9]oncontactgeometry. J.Martinetprovedthatevery
orientable 3-manifold admits a contact form [16]. In 1993, H. Geiges showed that
contact structures exist on simply connected 5-manifolds by applying results of
Y. Eliashberg and A. Weinstein on contact surgery [8]. However, little is known
about the existence of contact structures in high-dimensional manifolds. Since our
objectofinterestisasmoothseven-dimensionalmanifold, weconcernedChapter3
with G -geometry, which is the geometry of smooth seven-dimensional manifolds
2
with an additional structure and we refer to [3] and [12] for more details.
InChapter4,weprovethatanyseven-dimensionalmanifoldwithaG -structure
2
admits an almost contact structure, which follows Theorem 2. Additionally, we
constructexplicitlyanalmostcontactstructureonamanifoldwithaG -structure.
2
After discussing these results, in the remainder of the thesis, we return to
developing a description of how contact structures behave in manifolds with G -
2
structuresifcontactstructuresexist. Incontrastwithanalmostcontactstructure,
we first assume a contact structure exists on a manifold with a G -structure in
2
section 5.1. We define the notions of A-compatible and B-compatible contact
3
structures with a G -structure ϕ and prove Theorem 5, which states that this
2
definition of A-compatibility holds only on a manifold with a torsion free G -
2
structure and boundary because of Stokes’ theorem. After that, in section 5.2,
we introduce a new structure: a contact-G -structure which is also A-compatible
2
with ϕ on a seven-dimensional manifold. Most of the techniques discussed in
section 5.2 involve the decomposition of the space of 2-forms obtained from G -
2
representation.
We conclude the thesis in Chapter 6 by presenting examples with compatible
contact structures with ϕ on seven-dimensional manifolds.
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2 Almost contact structures and
Contact structures
This chapter is about the geometric background we need: we define an almost
contact structure and present some useful properties of it as well as a metric
associated with it. The second part not only defines contact structures and Reeb
vector fields but also contains a contact structure of S3 as an example. After
that, we conclude this chapter by providing how an almost contact structure can
be induced from a given contact structure. Throughout this chapter we assume
M is a smooth manifold with odd dimension. See [2] and [19] on the geometry of
almost contact manifolds.
2.1 Almost contact structures
Let M be a (2n + 1)-dimensional differential manifold and φ,ξ,η be a field of
endomorphisms of the tangent spaces TM as a (1,1)-tensor field, a vector field
and a 1-form on M respectively. If a triple (φ,ξ,η) satisfies two conditions
η(ξ) = 1 (2.1.1)
φ2(X) = −X +η(X)ξ (2.1.2)
for any vector field X on M, (φ,ξ,η) is called an almost contact structure and M
is called an almost contact manifold.