scholarly journals Contact Structures of Sasaki Type and Their Associated Moduli

2019 ◽  
Vol 6 (1) ◽  
pp. 1-30
Author(s):  
Charles P. Boyer
Keyword(s):  
Scalar Curvature ◽  
Present Article ◽  
Einstein Metrics ◽  
Contact Structure ◽  
Constant Scalar Curvature ◽  
Contact Geometry ◽  
Contact Structures ◽  
Sasakian Geometry ◽  
Sasakian Structures

Abstract This article is based on a talk at the RIEMain in Contact conference in Cagliari, Italy in honor of the 78th birthday of David Blair one of the founders of modern Riemannian contact geometry. The present article is a survey of a special type of Riemannian contact structure known as Sasakian geometry. An ultimate goal of this survey is to understand the moduli of classes of Sasakian structures as well as the moduli of extremal and constant scalar curvature Sasaki metrics, and in particular the moduli of Sasaki-Einstein metrics.

scholarly journals Brieskorn manifolds, positive Sasakian geometry, and contact topology

2016 ◽  
Vol 28 (5) ◽  
pp. 943-965
Author(s):  
Charles P. Boyer ◽  
Leonardo Macarini ◽  
Otto van Koert
Keyword(s):  
Moduli Space ◽  
Einstein Metrics ◽  
Contact Structures ◽  
Rational Homology ◽  
Contact Topology ◽  
Connected Sums ◽  
New Family ◽  
Rational Homology Spheres ◽  
Sasakian Geometry ◽  
Sasakian Structures

AbstractUsing ${S^{1}}$-equivariant symplectic homology, in particular its mean Euler characteristic, of the natural filling of links of Brieskorn–Pham polynomials, we prove the existence of infinitely many inequivalent contact structures on various manifolds, including in dimension 5 the k-fold connected sums of ${S^{2}\times S^{3}}$ and certain rational homology spheres. We then apply our result to show that on these manifolds the moduli space of classes of positive Sasakian structures has infinitely many components. We also apply our results to give lower bounds on the number of components of the moduli space of Sasaki–Einstein metrics on certain homotopy spheres. Finally, a new family of Sasaki–Einstein metrics of real dimension 20 on ${S^{5}}$ is exhibited.

scholarly journals A Survey of Riemannian Contact Geometry

2019 ◽  
Vol 6 (1) ◽  
pp. 31-64 ◽  
Author(s):  
David E. Blair
Keyword(s):  
Contact Structure ◽  
Positive Curvature ◽  
Contact Geometry ◽  
Contact Structures ◽  
Contact Manifolds ◽  
3 Dimensional ◽  
Contact Metric Manifolds ◽  
The Subject ◽  
Mere Existence ◽  
Curvature Functionals

AbstractThis survey is a presentation of the five lectures on Riemannian contact geometry that the author gave at the conference “RIEMain in Contact”, 18-22 June 2018 in Cagliari, Sardinia. The author was particularly pleased to be asked to give this presentation and appreciated the organizers’ kindness in dedicating the conference to him. Georges Reeb once made the comment that the mere existence of a contact form on a manifold should in some sense “tighten up” the manifold. The statement seemed quite pertinent for a conference that brought together both geometers and topologists working on contact manifolds, whether in terms of “tight” vs. “overtwisted” or whether an associated metric should have some positive curvature. The first section will lay down the basic definitions and examples of the subject of contact metric manifolds. The second section will be a continuation of the first discussing tangent sphere bundles, contact structures on 3-dimensional Lie groups and a brief treatment of submanifolds. Section III will be devoted to the curvature of contact metric manifolds. Section IV will discuss complex contact manifolds and some older style topology. Section V treats curvature functionals and Ricci solitons. A sixth section has been added giving a discussion of the question of whether a Riemannian metric g can be an associated metric for more than one contact structure; at the conference this was an addendum to the third lecture.

scholarly journals Critical associated metrics on contact manifolds. II

1986 ◽  
Vol 41 (3) ◽  
pp. 404-410 ◽  
Author(s):  
D. E. Blair ◽  
A. J. Ledger
Keyword(s):  
Scalar Curvature ◽  
Critical Points ◽  
Compact Manifold ◽  
Einstein Metrics ◽  
Contact Structure ◽  
Riemannian Metrics ◽  
Contact Manifold ◽  
Contact Manifolds ◽  
Contact Metric Structure ◽  
Sasakian Metrics

AbstractThe study of the integral of the scalar curvature, ∫MRdVg, as a function on the set of all Riemannian metrics of the same total volume on a compact manifold is now classical, and the critical points are the Einstein metrics. On a compact contact manifold we consider this and ∫M (R − R* − 4n2) dv, with R* the *-scalar curvature, as functions on the set of metrics associated to the contact structure. For these integrals the critical point conditions then become certain commutativity conditions on the Ricci operator and the fundamental collineation of the contact metric structure. In particular, Sasakian metrics, when they exist, are maxima for the second function.

scholarly journals Generalizations of Kähler-Ricci solitons on projective bundles

2011 ◽  
Vol 108 (2) ◽  
pp. 161 ◽  
Author(s):  
Gideon Maschler ◽  
Christina W. Tønnesen-Friedman
Keyword(s):  
Scalar Curvature ◽  
Existence Theorem ◽  
Einstein Metrics ◽  
Constant Scalar Curvature ◽  
Ricci Solitons ◽  
Kähler Metrics ◽  
Kahler Metrics ◽  
Projective Bundles ◽  
Affine Combination ◽  
Constant Scalar Curvature Metrics

We prove that an admissible manifold (as defined by Apostolov, Calderbank, Gauduchon and Tønnesen-Friedman), arising from a base with a local Kähler product of constant scalar curvature metrics, admits Generalized Quasi-Einstein Kähler metrics (as defined by D. Guan) in all "sufficiently small" admissible Kähler classes. We give an example where the existence of Generalized Quasi-Einstein metrics fails in some Kähler classes while not in others. We also prove an analogous existence theorem for an additional metric type, defined by the requirement that the scalar curvature is an affine combination of a Killing potential and its Laplacian.

scholarly journals Existence and non-uniqueness of constant scalar curvature toric Sasaki metrics

2011 ◽  
Vol 147 (5) ◽  
pp. 1613-1634 ◽  
Author(s):  
Eveline Legendre
Keyword(s):  
Finite Number ◽  
Scalar Curvature ◽  
Vector Fields ◽  
Existence Result ◽  
Constant Scalar Curvature ◽  
Contact Structures ◽  
Toric Geometry ◽  
Contact Manifolds ◽  
Reeb Vector ◽  
Toric Contact Manifolds

AbstractWe study compatible toric Sasaki metrics with constant scalar curvature on co-oriented compact toric contact manifolds of Reeb type of dimension at least five. These metrics come in rays of transversal homothety due to the possible rescaling of the Reeb vector fields. We prove that there exist Reeb vector fields for which the transversal Futaki invariant (restricted to the Lie algebra of the torus) vanishes. Using an existence result of E. Legendre [Toric geometry of convex quadrilaterals, J. Symplectic Geom. 9 (2011), 343–385], we show that a co-oriented compact toric contact 5-manifold whose moment cone has four facets admits a finite number of rays of transversal homothetic compatible toric Sasaki metrics with constant scalar curvature. We point out a family of well-known toric contact structures on S2×S3 admitting two non-isometric and non-transversally homothetic compatible toric Sasaki metrics with constant scalar curvature.

b-Stability and Blow-Ups

2013 ◽  
Vol 57 (1) ◽  
pp. 125-137 ◽  
Author(s):  
S. K. Donaldson
Keyword(s):  
Scalar Curvature ◽  
Einstein Metrics ◽  
Constant Scalar Curvature ◽  
Birational Transformations ◽  
Definition Of ◽  
Kähler Einstein Metrics

AbstractWe study a notion of ‘b-stability’, introduced previously by the author in connection with the existence of constant scalar curvature Kähler, and Kähler-Einstein, metrics. The main result is Theorem 1.2, which makes progress towards a statement that the existence of such metrics implies b-stability. The proof is a modification of an argument of Stoppa, taking account of the birational transformations involved in the definition of b-stability.

scholarly journals Extremal Sasakian geometry on and related manifolds

2013 ◽  
Vol 149 (8) ◽  
pp. 1431-1456 ◽  
Author(s):  
Charles P. Boyer ◽  
Christina W. Tønnesen-Friedman
Keyword(s):  
Infinite Number ◽  
Contact Structures ◽  
Extremal Metrics ◽  
Sasakian Geometry ◽  
Sasakian Structures ◽  
Countably Infinite

AbstractWe prove the existence of extremal Sasakian structures occurring on a countably infinite number of distinct contact structures on${T}^{2} \times {S}^{3} $and certain related 5-manifolds. These structures occur in bouquets and exhaust the Sasaki cones in all except one case in which there are no extremal metrics.

scholarly journals FANO MANIFOLDS, CONTACT STRUCTURES, AND QUATERNIONIC GEOMETRY

1995 ◽  
Vol 06 (03) ◽  
pp. 419-437 ◽  
Author(s):  
CLAUDE LEBRUN
Keyword(s):  
Riemannian Manifold ◽  
Scalar Curvature ◽  
Contact Structure ◽  
Twistor Space ◽  
Positive Scalar Curvature ◽  
Einstein Metric ◽  
Contact Structures ◽  
Fano Manifolds ◽  
Quaternionic Geometry ◽  
Structure Formula

Let Z be a compact complex (2n+1)-manifold which carries a complex contact structure, meaning a codimension-1 holomorphic sub-bundle D⊂TZ which is maximally non-integrable. If Z admits a Kähler-Einstein metric of positive scalar curvature, we show that it is the Salamon twistor space of a quaternion-Kähler manifold (M4n, g). If Z also admits a second complex contact structure [Formula: see text], then Z=CP2n+1. As an application, we give several new characterizations of the Riemannian manifold HPn= Sp(n+1)/(Sp(n)×Sp(1)).

scholarly journals Warped product Einstein metrics over spaces with constant scalar curvature

2014 ◽  
Vol 18 (1) ◽  
pp. 159-190 ◽  
Author(s):  
Chenxu He ◽  
Peter Petersen ◽  
William Wylie
Keyword(s):  
Scalar Curvature ◽  
Warped Product ◽  
Einstein Metrics ◽  
Constant Scalar Curvature
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