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2002 | Buch

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Riemannian Geometry of Contact and Symplectic Manifolds

verfasst von: David E. Blair

Verlag: Birkhäuser Boston

Buchreihe : Progress in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

The author's lectures, "Contact Manifolds in Riemannian Geometry," volume 509 (1976), in the Springer-Verlag Lecture Notes in Mathematics series have been out of print for some time and it seems appropriate that an expanded version of this material should become available. The present text deals with the Riemannian geometry of both symplectic and contact manifolds, although the book is more contact than symplectic. This work is based on the recent research of the author, his students, colleagues, and other scholars, the author's graduate courses at Michigan State University and the earlier lecture notes. Chapter 1 presents the general theory of symplectic manifolds. Principal circle bundles are then discussed in Chapter 2 as a prelude to the Boothby Wang fibration of a compact regular contact manifold in Chapter 3, which deals with the general theory of contact manifolds. Chapter 4 focuses on Rie mannian metrics associated to symplectic and contact structures. Chapter 5 is devoted to integral submanifolds of the contact subbundle. In Chapter 6 we discuss the normality of almost contact structures, Sasakian manifolds, K contact manifolds, the relation of contact metric structures and CR-structures, and cosymplectic structures. Chapter 7 deals with the important study of the curvature of a contact metric manifold. In Chapter 8 we give a selection of results on submanifolds of Kahler and Sasakian manifolds, including an illus tration of the technique of A. Ros in a theorem of F. Urbano on compact minimal Lagrangian sub manifolds in cpn.

Inhaltsverzeichnis

Frontmatter

1. Symplectic Manifolds

Abstract

To set the stage for our development, we begin this book with a treatment of the basic features of symplectic geometry. By a symplectic manifold we mean an even-dimensional differentiable (C ^∞) manifold M ²ⁿn together with a global 2-form Ω which is closed and of maximal rank, i.e., dΩ = 0, Ωⁿ ≠ 0. By a symplectomorphism f: (M ₁, Ω₁) → (M ₂, Ω₂) we mean a diffeomorphism f : M ₁ → M ₂ such that f*Ω₂ =Ω₁.

David E. Blair

2. Principal S 1-bundles

Let P and M be C ^∞ manifolds, π : P → M a C ^∞ map of P onto M and G a Lie group acting on P to the right. Then (P, G, M) is called a principal G-bundle if

G acts freely on P,

π(p ₁) = π(p ₂) if and only if there exists g ∈ G such that p ₁ g = p ₂,

P is locally trivial over M, i.e., for every m ∈ M there exists a neighborhood U of m and a map F _u : π_-1(U) → G such that F _u(pg) = (F _u(p))g and such that the map Ψ : π_-1(U) → U × G taking p to (π(p), F _u(p)) is a diffeomorphism.

David E. Blair

3. Contact Manifolds

Abstract

By a contact manifold we mean a C ^∞ manifold M ²ⁿ⁺¹ together with a 1-form η such that η ∧ (dη)ⁿ ≠ 0. In particular η ∧ (dη)ⁿ ≠ 0 is a volume element on M so that a contact manifold is orientable. Also dη has rank 2n on the Grassmann algebra ∧ T _m ^* M at each point m ∈ M and thus we have a 1-dimensional subspace, {X ∈ T _m M|dη(X, T _m M) = 0}, on which η ≠ 0 and which is complementary to the subspace on which η = 0. Therefore choosing ξ _m in this subspace normalized by η(ξ _m) = 1 we have a global vector field ξ satisfying

$$ d\eta \left( {\xi ,X} \right) = 0,\;\eta \left( \xi \right) = 1 $$

David E. Blair

4. Associated Metrics

Abstract

We will generally regard the theory of almost Hermitian structures as well known and give here only definitions and a few properties that will be important for our study; many of these were already mentioned in Chapter 1. For more detail the reader is referred to Gray and Hervella [1980] , Kobayashi-Nomizu [1963–69, Chapter IX] and Kobayashi-Wu [1983]; also, despite its classical nature, the book of Yano [1965] contains helpful information on many of these structures.

David E. Blair

5. Integral Submanifolds and Contact Transformations

Abstract

Let M ²ⁿ⁺¹ be a contact manifold with contact form η. We have seen that η = 0 defines a 2n-dimensional subbundle D called the contact distribution or subbundle and that since η ∧ (dη)n ≠ 0, D is non-integrable. This nonintegrability was easily visualized, for example, in Example 3.2.6.

David E. Blair

6. Sasakian and Cosymplectic Manifolds

Abstract

Recall that almost contact manifolds were defined as manifolds with structural group U(n) × 1 and hence can be thought of as odd-dimensional analogues of almost complex manifolds. We now consider almost contact manifolds which are, in the sense to be defined, analogous to compiex manifolds.

David E. Blair

7. Curvature of Contact Metric Manifolds

Abstract

In this chapter we discuss many aspects of the curvature of contact metric manifolds. We begin with some preliminaries concerning the tensor field h. Let M ²ⁿ⁺¹ be a contact metric manifold with structure tensors (ϕ, ξ, η, g) and h = ½£ _ξ ϕ as before. Recall that in Lemma 6.2 we saw that ∇_X ξ = -ϕX — ϕhX.

David E. Blair

8. Submanifolds of Kähler and Sasakian Manifolds

Abstract

In this chapter we study submanifolds in both contact and Kähler geometry. These are extensive subjects in their own right and we give only a few basic results. For a submanifold M of a Riemannian manifold (M̄, g̃) we denote the induced metric by g. Then the Levi-Cività connection ∇ of g and the second fundamental form σ are related to the ambient Levi-Cività connection ∇̃ by

$${\bar\nabla_X}Y = {\nabla_X}Y +\sigma(X,Y)$$

David E. Blair

9. Tangent Bundles and Tangent Sphere Bundles

Abstract

In the first two sections of this chapter we discuss the geometry of the tangent bundle and the tangent sphere bundle. In Section 3 we briefly present a more general construction on vector bundles and in Section 4 specialize to the case of the normal bundle of a submanifold. The formalism for the tangent bundle and the tangent sphere bundle is of sufficient importance to warrant its own development, rather than specializing from the vector bundle case. As we saw in Chapter 1, the cotangent bundle of a manifold has a natural symplectic structure and we will see here that the same is true of the tangent bundle of a Riemannian manifold.

David E. Blair

10. Curvature Functionals on Spaces of Associated Metrics

Abstract

The study of the integral of the scalar curvature, A(g) = ∫_M τ dV _g, as a functional on the set M ₁ of all Riemannian metrics of the same total volume on a compact orientable manifold M is now classical, dating back to Hilbert [1915] (see also Nagano [1967]). A Riemannian metric g is a critical point of A(g) if and only if g is an Einstein metric. Since there are so many Riemannian metrics on a manifold, one can regard, philosophically, the finding of critical metrics as an approach to searching for the best metric for the given manifold. Other functions of the curvature have been taken as integrands as well, most notably $ B(g) = {\int_M {{\tau ^2}d{V_g},\;C(g) = \int_M {\left| \rho \right|} } ^2}d{V_g} $ where ρ is the Ricci tensor, and $D(g) = \int_M {{{\left| {{R_{kjih}}} \right|}^2}} d{V_g}$; the critical point conditions for these have been computed by Berger [1970]. From the critical point conditions it is easy to see that Einstein metrics are critical for B(g) and C(g) but not necessarily conversely. For example an η-Einstein manifold M ²ⁿ⁺¹ with scalar curvature equal to 2n(2n + 1) or 2n(2n + 3) is a non-Einstein critical metric of C(g), Yamaguchi and Chūman [1983]. In the case of B(g) Yamaguchi and Chūman showed that a Sasakian critical point is Einstein. Similarly metrics of constant curvature and Kähler metrics of constant holomorphic curvature are critical for D(g), see Muto [1975]; also a Sasakian manifold of dimension m and constant ϕ-sectional curvature 3m — 1 is critical for D(g), see Yamaguchi and Chūman [1983].

David E. Blair

11. Negative ξ-sectional Curvature

Abstract

The purpose of this chapter is to introduce some special directions that belong to the contact subbundle of a contact metric manifold with negative sectional curvature for plane sections containing the characteristic vector field ξ or more generally when the operator h admits an eigenvalue greater than 1; these directions were introduced by the author in [1998]. We also discuss in this chapter some questions concerning Anosov and conformally Anosov flows. For simplicity we will often refer to the sectional curvature of plane sections containing the characteristic vector field ξ as ξ-sectional curvature.

David E. Blair

12. Complex Contact Manifolds

Abstract

While the study of complex contact manifolds is almost as old as the modern theory of real contact manifolds, the subject has received much less attention and as many examples are now appearing in the literature, especially twistor spaces over quaternionic Kähler manifolds (e.g., LeBrun [1991], [1995], Moroianu and Semmelmann [1994], Salamon [1982], Ye [1994]), the time is ripe for another look at the subject. As an indication of this interest we note, for example, the following result of Moroianu and Semmelmann [1994] that on a compact spin Kähler manifold M of positive scalar curvature and complex dimension 4l + 3, the following are equivalent: (i) M is a Kähler—Einstein manifold admitting a complex contact structure, (ii) M is the twistor space of a quaternionic Kähler manifold of positive scalar curvature, (iii) M admits Kählerian Killing spinors. LeBrun [1995] proves that a complex contact manifold of positive first Chern class, i.e., a Fano contact manifold, is a twistor space if and only if it admits a Kähler-Einstein metric and conjectures that every Fano contact manifold is a twistor space.

David E. Blair

13. 3-Sasakian Manifolds

Abstract

As with the last chapter we will give more of a survey and only a few proofs. Another survey of both history and recent work on 3-Sasakian manifolds is Boyer and Galicki [1999].

David E. Blair

Backmatter

Titel: Riemannian Geometry of Contact and Symplectic Manifolds
verfasst von: David E. Blair
Verlag: Birkhäuser Boston
Electronic ISBN: 978-1-4757-3604-5
Print ISBN: 978-1-4757-3606-9
DOI: https://doi.org/10.1007/978-1-4757-3604-5

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Über dieses Buch

Inhaltsverzeichnis

Frontmatter

1. Symplectic Manifolds

2. Principal S 1-bundles

3. Contact Manifolds

4. Associated Metrics

5. Integral Submanifolds and Contact Transformations

6. Sasakian and Cosymplectic Manifolds

7. Curvature of Contact Metric Manifolds

8. Submanifolds of Kähler and Sasakian Manifolds

9. Tangent Bundles and Tangent Sphere Bundles

10. Curvature Functionals on Spaces of Associated Metrics

11. Negative ξ-sectional Curvature

12. Complex Contact Manifolds

13. 3-Sasakian Manifolds

Backmatter