المؤلفون: Boyer, Charles P., Galicki, Krzysztof, Simanca, Santiago R.

المصدر: Proceedings of the Conference on Riemannian Topology, pg 263-290, K. Galicki & S. Simanca, Eds, Birkhauser, Boston, 2008.

مصطلحات موضوعية: Mathematics - Differential Geometry, 53C25

الوصف: We study the Sasaki cone of a CR structure of Sasaki type on a given closed manifold. We introduce an energy functional over the cone, and use its critical points to single out the strongly extremal Reeb vectors fields. Should one such vector field be a member of the extremal set, the scalar curvature of a Sasaki extremal metric representing it would have the smallest $L^2$-norm among all Sasakian metrics of fixed volume that can represent vector fields in the cone. We use links of isolated hypersurface singularities to produce examples of manifolds of Sasaki type, many of these in dimension five, whose Sasaki cone coincides with the extremal set, and examples where the extremal set is empty. We end up by proving that a conjecture of Orlik concerning the torsion of the homology groups of these links holds in the five dimensional case.
Comment: 24 pages, to appear in the Proceedings of the Conference on Riemannian Topology, K. Galicki and S.R. Simanca Eds., Birkhauser, Boston

الوصول الحر: http://arxiv.org/abs/0801.0217Test

المؤلفون: Boyer, Charles P., Galicki, Krzysztof

مصطلحات موضوعية: Mathematics - Differential Geometry, High Energy Physics - Theory, Mathematical Physics, 53C25

الوصف: In this expository article we discuss the relations between Sasakian geometry, reduced holonomy and supersymmetry. It is well known that the Riemannian manifolds other than the round spheres that admit real Killing spinors are precisely Sasaki-Einstein manifolds, 7-manifolds with a nearly parallel G2 structure, and nearly Kaehler 6-manifolds. We then discuss the relations between the latter two and Sasaki-Einstein geometry.
Comment: 40 pages, some minor corrections made, to appear in the Handbook of pseudo-Riemannian Geometry and Supersymmetry

الوصول الحر: http://arxiv.org/abs/math/0703231Test

المؤلفون: Boyer, Charles P., Galicki, Krzysztof, Simanca, Santiago R.

المصدر: Commun.Math.Phys.279:705-733,2008

مصطلحات موضوعية: Mathematics - Differential Geometry, 53C25

الوصف: Let $M$ be a closed manifold of Sasaki type. A polarization of $M$ is defined by a Reeb vector field, and for one such, we consider the set of all Sasakian metrics compatible with it. On this space, we study the functional given by the squared $L^2$-norm of the scalar curvature. We prove that its critical points, or canonical representatives of the polarization, are Sasakian metrics that are transversally extremal. We define a Sasaki-Futaki invariant of the polarization, and show that it obstructs the existence of constant scalar curvature representatives. For a fixed CR structure of Sasaki type, we define the Sasaki cone of structures compatible with this underlying CR structure, and prove that the set of polarizations in it that admit a canonical representative is open.
Comment: 36 pages, minor corrections made, example added

الوصول الحر: http://arxiv.org/abs/math/0604325Test

المؤلفون: Boyer, Charles P., Galicki, Krzysztof, Ornea, Liviu

المصدر: Mathematische Zeitschrift 257 (2007), 907-924.

مصطلحات موضوعية: Mathematics - Differential Geometry, Mathematics - Symplectic Geometry, 53C25

الوصف: We describe various constructions in Sasakian geometry. First we generalize the join construction of the first two authors to arbitrary Sasakian manifolds. We then give several examples, including ones which prove the existence of Sasakian-Einstein metrics on manifolds homeomorphic to $S^2\times S^5.$ Then we use a generalization of the join construction due to Lerman, namely contact fibre bundles, to give a theorem constructing toric Sasakian structures. Finally, we explicitly construct regular toric Sasakian structures on all simply connected regular contact manifolds in dimension five.
Comment: Some minor errors corrected. References updated. To appear in Mathematische Zeitschrift

الوصول الحر: http://arxiv.org/abs/math/0602233Test

المؤلفون: Boyer, Charles P, Galicki, Krzysztof

المصدر: Geom. Topol. 10 (2006) 2219-2235

مصطلحات موضوعية: Mathematics - Differential Geometry, Mathematics - Algebraic Topology, 53C25, 57R55

الوصف: We prove the existence of Sasakian metrics with positive Ricci curvature on certain highly connected odd dimensional manifolds. In particular, we show that manifolds homeomorphic to the 2k-fold connected sum of S^{2n-1} x S^{2n} admit Sasakian metrics with positive Ricci curvature for all k. Furthermore, a formula for computing the diffeomorphism types is given and tables are presented for dimensions 7 and 11.
Comment: This is the version published by Geometry & Topology on 29 November 2006

الوصول الحر: http://arxiv.org/abs/math/0508189Test

المؤلفون: Boyer, Charles P., Galicki, Krzysztof

المصدر: Perspectives in Riemannian geometry, CRM Proc. Lecture Notes, 40, 2006, 47-61.

مصطلحات موضوعية: Mathematics - Differential Geometry, Mathematics - Algebraic Geometry

الوصف: This paper is based on a talk presented by the first author at the Short Program on Riemannian Geometry that took place at the Centre de Recherche Math\'ematiques, Universit\'e de Montr\'eal, during the period June 28-July 16, 2004. It is a report on our joint work with J\'anos Koll\'ar concerning the existence of an abundance of Einstein metrics on odd dimensional spheres, including exotic spheres.
Comment: 14 pages, Expository paper to appear in the Proceedings of the Short Program on Riemannian Geometry

الوصول الحر: http://arxiv.org/abs/math/0505221Test

المؤلفون: Boyer, Charles P., Galicki, Krzysztof, Matzeu, Paola

المصدر: Commun.Math.Phys.262:177-208,2006

مصطلحات موضوعية: Mathematics - Differential Geometry, High Energy Physics - Theory

الوصف: We study eta-Einstein geometry as a class of distinguished Riemannian metrics on contact metric manifolds. In particular, we use a previous solution of the Calabi problem for Sasakian geometry to prove the existence of eta-Einstein structures on many different compact manifolds, including exotic spheres. We also relate these results to the existence of Einstein-Weyl structures.
Comment: 31 pages, minor changes made, to appear in Commun. Math. Phys

الوصول الحر: http://arxiv.org/abs/math/0406627Test

المؤلفون: Boyer, Charles P., Galicki, Krzysztof

المصدر: Supplemento ai Rendiconti del Circolo Matematico di Palermo Serie II. Suppl 75 (2005), 57-87.

مصطلحات موضوعية: Mathematics - Differential Geometry, High Energy Physics - Theory, Mathematics - Algebraic Geometry, 53C25

الوصف: We review our study of Sasakian geometry as an agent for proving the existence of Einstein metrics on odd dimensional manifolds. Particular emphasis is given to the Sasakian structures occuring on links of isolated hypersurface singularities.
Comment: This is an expository article that grew out of notes for the three lectures the second author presented during the XXIV-th Winter School of {\it Geometry and Physics} in Srni, Czech Republic, in January of 2004., 30 pages. Some new examples and references were added

الوصول الحر: http://arxiv.org/abs/math/0405256Test

المؤلفون: Boyer, Charles P., Galicki, Krzysztof

المصدر: J.Diff.Geom.74:353-362,2006

مصطلحات موضوعية: Mathematics - Differential Geometry, High Energy Physics - Theory, Mathematics - Algebraic Geometry, 53C20,53C12,14E30

الوصف: We prove the existence of Sasakian-Einstein metrics on infinitely many rational homology spheres in all odd dimensions greater than 3. In dimension 5 we obain somewhat sharper results. There are examples where the number of effective parameters in the Einstein metric grows exponentially with dimension.
Comment: 8 pages

الوصول الحر: http://arxiv.org/abs/math/0311355Test

المؤلفون: Boyer, Charles P., Galicki, Krzysztof, Kollár, János, Thomas, Evan

المصدر: Experimental Mathematics 14 (2005), 61-66.

مصطلحات موضوعية: Mathematics - Differential Geometry, High Energy Physics - Theory, Mathematics - Algebraic Geometry, 53C20, 53C12, 14E30

الوصف: In a recent article the first three authors proved that in dimension $4m+1$ all homotopy spheres that bound parallelizable manifolds admit Einstein metrics of positive scalar curvature which, in fact, are Sasakian-Einstein. They also conjectured that all such homotopy spheres in dimension $4m-1, m\geq2$ admit Sasakian-Einstein metrics \cite{BGK}, and proved this for the simplest case, namely dimension $7.$ In this paper we describe computer programs that show that this conjecture is also true for 11-spheres and 15-spheres. Moreover, a program is given that determines the partition of the 8610 deformation classes of Sasakian-Einstein metrics into the 28 distinct oriented diffomorphism types in dimension $7.$
Comment: 7 pages

الوصول الحر: http://arxiv.org/abs/math/0311293Test