Showing papers by "David E. Blair published in 2002"

Riemannian Geometry of Contact and Symplectic Manifolds

Abstract: Preface * 1. Symplectic Manifolds * 2. Principal S1-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 Kahler and Sasakian Manifolds * 9. Tangent Bundles and Tangent Sphere Bundles * 10. Curvature Functionals and Spaces of Associated Metrics * 11. Negative Xi-sectional Curvature * 12. Complex Contact Manifolds * 13. Additional Topics in Complex Geometry * 14. 3-Sasakian Manifolds * Bibliography * Subject Index * Author Index

Cr products in locally conformal kahler manifolds

Abstract: We study CR products (in the sense of Chen (J. Diff. Geom. 16 (1981), 305-322, 493-509)) in locally conformal Kahler (l.c.K.) manifolds. We show that a CR submanifold M m of a l.c.K. manifold has a parallel f -structure P if and only if it is a restricted CR product (i.e. both the holomorphic and totally real distributions D and D ⊥ are parallel and D has complex dimension 1 whenever M m is not orthogonal to the Lee field). We study rough CR products, i.e. CR submanifolds in a l.c.K. manifold whose local CR manifolds {Mi}i∈I are CR products (relative to the local Kahler metrics {˜ gi}i∈I of the ambient space). If M m is a standard rough CR product of a complex Hopf manifold, each leaf of the Levi foliation, orthogonal to the Lee field, is shown to be isometric to the sphere S 2 . Any warped product CR submanifold M m = M ⊥ ×f M T , with M ⊥ anti-invariant and M T invariant is shown to be a CR product, provided that the tangential component of the Lee field (of the ambient l.c.K. manifold) is orthogonal to M ⊥ .

Pseudohermitian geometry on contact Riemannian manifolds

Abstract: Starting from work by S. Tanno, [39], and E. Barletta et al., [3], we study the geometry of (possibly non integrable) almost CR structures on contact Riemannian manifolds. We characterize CR-pluriharmonic functions in terms of differential operators naturally attached to the given contact Riemannian structure. We show that the almost CR structure of a contact Riemannian manifold (M, η) admitting global 276 DAVID E. BLAIR – SORIN DRAGOMIR [2] nonzero closed sections (with respect to which η is volume normalized) in the canonical bundle is integrable and η is a pseudo-Einstein contact form. The pseudohermitian holonomy of a Sasakian manifold M is shown to be contained in SU(n)× 1 if and only if the Tanaka-Webster connection is Ricci flat. Also, for any quaternionic Sasakian manifold (M, (F, T, θ, g)) either the Tanaka-Webster connection of (M, θ) is Ricci flat or m = 1 and then (M, θ) is pseudo-Einstein if and only if 4p + ρ∗ θ is closed, where p is a local 1-form on M such that ∇G = p ⊗ H and ∇H = −p ⊗ G for some frame {F, G, H}, and ρ∗ is the pseudohermitian scalar curvature of (M, θ). On any Sasakian manifold M there is a smooth integrable Pfaffian system, invariant by Ψ(x) (the pseudohermitian holonomy group at x ∈ M) containing the contact flow as a subfoliation. We build canonical connections (reminiscent of the Tanaka connection, [38]) on complex vector bundles over contact Riemannian manifolds, carrying a pre-∂-operator and a Hermitian metric. As an application, we compute the first structure function of the underlying almost CR structure of a contact Riemannian manifold. The restricted conformal class [Gη] of the (generalized) Fefferman metric and certain canonical connections D (with trace ΛgR D = 0) are shown to be gauge invariants.