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47
On EtaEinstein Sasakian Geometry
 Comm. Math. Phys
"... The purpose of this paper is to study a special kind of Riemannian metrics on Sasakian manifolds. A Sasakian manifold M of dimension 2n + 1 with a Sasakian structure S = (ξ, η,Φ, g) is said to be ηEinstein if the Ricci curvature tensor of the metric g satisfies the equation Ricg = λg + νη ⊗ η for s ..."
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Cited by 34 (9 self)
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The purpose of this paper is to study a special kind of Riemannian metrics on Sasakian manifolds. A Sasakian manifold M of dimension 2n + 1 with a Sasakian structure S = (ξ, η,Φ, g) is said to be ηEinstein if the Ricci curvature tensor of the metric g satisfies the equation Ricg = λg + νη ⊗ η for some constants λ, ν ∈ R.
The EinsteinDirac Equation on Riemannian Spin Manifolds.
 Journ. Geom. Phys
, 1999
"... We construct exact solutions of the EinsteinDirac equation, which couples the gravitational field with an eigenspinor of the Dirac operator via the energymomentum tensor. For this purpose we introduce a new field equation generalizing the notion of Killing spinors. The solutions of this spinor fiel ..."
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Cited by 23 (0 self)
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We construct exact solutions of the EinsteinDirac equation, which couples the gravitational field with an eigenspinor of the Dirac operator via the energymomentum tensor. For this purpose we introduce a new field equation generalizing the notion of Killing spinors. The solutions of this spinor field equation are called weak Killing spinors (WKspinors). They are special solutions of the EinsteinDirac equation and in dimension n = 3 the two equations essentially coincide. It turns out that any Sasakian manifold with Ricci tensor related in some special way to the metric tensor as well as to the contact structure admits a WKspinor. This result is a consequence of the investigation of special spinor field equations on Sasakian manifolds (Sasakian quasiKilling spinors). Altogether, in odd dimensions a contact geometry generates a solution of the EinsteinDirac equation. Moreover, we prove the existence of solutions of the EinsteinDirac equations that are not WKspinors in all dimension...
New Einstein Metrics in Dimension Five
, 2001
"... The purpose of this note is to prove the existence of new SasakianEinstein met rics on S 2 x S a and on (S 2 x Sa)#(S = x Sa). These give the first known examples of nonregular SasakianEinstein 5manifolds. Our method involves describing the SasakianEinstein structures as links of certain is ..."
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Cited by 20 (10 self)
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The purpose of this note is to prove the existence of new SasakianEinstein met rics on S 2 x S a and on (S 2 x Sa)#(S = x Sa). These give the first known examples of nonregular SasakianEinstein 5manifolds. Our method involves describing the SasakianEinstein structures as links of certain isolated hypersurface singularities, and makes use of the recent work of Demailly and Kolltr who obtained new examples of KiihlerEinstein del Pezzo surfaces with quotient singularities.
Einstein metrics on spheres
 Ann. of Math
, 2005
"... Any sphere S n admits a metric of constant sectional curvature. These canonical metrics are homogeneous and Einstein, that is the Ricci curvature is a constant multiple of the metric. The spheres S 4m+3, m> 1 are known to have another Sp(m + 1)homogeneous Einstein metric discovered by Jensen [Jen73 ..."
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Cited by 18 (13 self)
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Any sphere S n admits a metric of constant sectional curvature. These canonical metrics are homogeneous and Einstein, that is the Ricci curvature is a constant multiple of the metric. The spheres S 4m+3, m> 1 are known to have another Sp(m + 1)homogeneous Einstein metric discovered by Jensen [Jen73]. In addition,
Sasakian geometry, hypersurface singularities, and Einstein
, 2005
"... This review article has grown out of notes for the three lectures the second author presented during the XXIVth Winter School of Geometry and Physics in Srni, Czech Republic, in January of 2004. Our purpose is twofold. We want give a brief introduction to some of the techniques we have developed ov ..."
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Cited by 12 (3 self)
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This review article has grown out of notes for the three lectures the second author presented during the XXIVth Winter School of Geometry and Physics in Srni, Czech Republic, in January of 2004. Our purpose is twofold. We want give a brief introduction to some of the techniques we have developed over the last 5 years
CalabiYau connections with torsion on toric bundles
"... We find sufficient conditions for principal toric bundles over compact Kähler manifolds to admit CalabiYau connections with torsion, as well as conditions to admit strong Kähler connections with torsion. With the aid of a topological classification, we construct such geometry on (k −1)(S 2 ×S 4)#k( ..."
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Cited by 10 (0 self)
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We find sufficient conditions for principal toric bundles over compact Kähler manifolds to admit CalabiYau connections with torsion, as well as conditions to admit strong Kähler connections with torsion. With the aid of a topological classification, we construct such geometry on (k −1)(S 2 ×S 4)#k(S 3 ×S 3) for all k ≥ 1. 1.
Sasakian geometry, homotopy spheres and positive Ricci curvature
, 2002
"... We discuss the Sasakian geometry of odd dimensional homotopy spheres. In particular, we give a completely new proof of the existence of metrics of positive Ricci curvature on exotic spheres that can be realized as the boundary of a parallelizable manifold. Furthermore, it is shown that on such hom ..."
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Cited by 10 (4 self)
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We discuss the Sasakian geometry of odd dimensional homotopy spheres. In particular, we give a completely new proof of the existence of metrics of positive Ricci curvature on exotic spheres that can be realized as the boundary of a parallelizable manifold. Furthermore, it is shown that on such homotopy spheres Σ 2n+1 the moduli space of Sasakian structures has infinitely many positive components determined by inequivalent underlying contact structures. We also prove the existence of Sasakian metrics with positive Ricci curvature on each of the 2 2m distinct diffeomorphism types of homotopy real projective spaces RP 4m+1.
Dibaryon spectroscopy
 JHEP
, 2003
"... The AdS/CFT correspondence relates dibaryons in superconformal gauge theories to holomorphic curves in KählerEinstein surfaces. The degree of the holomorphic curves is proportional to the gauge theory conformal dimension of the dibaryons. Moreover, the number of holomorphic curves should match, in ..."
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Cited by 10 (1 self)
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The AdS/CFT correspondence relates dibaryons in superconformal gauge theories to holomorphic curves in KählerEinstein surfaces. The degree of the holomorphic curves is proportional to the gauge theory conformal dimension of the dibaryons. Moreover, the number of holomorphic curves should match, in an appropriately defined sense, the number of dibaryons. Using AdS/CFT backgrounds built from the generalized conifolds of Gubser, Shatashvili, and Nekrasov (1999), we show that the gauge theory prediction for the dimension of dibaryonic operators does indeed match the degree of the corresponding holomorphic curves. For AdS/CFT backgrounds built from cones over del Pezzo surfaces, we are able to match the degree of the curves to the conformal dimension of dibaryons for the nth del Pezzo surface, 1 ≤ n ≤ 6. Also, for the del Pezzos and the Ak type generalized conifolds, for the dibaryons of smallest conformal dimension, we are able to match the number of holomorphic curves with the number of possible dibaryon operators from gauge theory. AdS/CFT correspondence [1, 2, 3] asserts that type IIB string theory on AdS5 × X is
EINSTEIN METRICS ON 5DIMENSIONAL SEIFERT BUNDLES
, 2008
"... The aim of this paper is to study the existence of positive Ricci curvature metrics, and especially Einstein metrics, on 5dimensional Seifert bundles. Seifert fibered 3–manifolds were introduced and studied in [Sei32]. These are 3–manifolds M which admit a differentiable map f: M → F to a surface F ..."
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Cited by 10 (1 self)
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The aim of this paper is to study the existence of positive Ricci curvature metrics, and especially Einstein metrics, on 5dimensional Seifert bundles. Seifert fibered 3–manifolds were introduced and studied in [Sei32]. These are 3–manifolds M which admit a differentiable map f: M → F to a surface F such that every fiber is a circle. There are finitely many points p1,...,pn ∈ F such that M is a circle bundle over F \ {p1,..., pn}, and the fibers over pi naturally appear with some multiplicty mi. It is the orbifold Euler charactersitic e orb (F, (p1, m1),...,(pn, mn)): = e(F) − ∑ (1 − 1 mi) that by and large determines the geometry of M. For instance, M has spherical geometry iff eorb> 0 and H1(M, Q) = 0 (see, for instance, [Sco83]). Higher dimensional Seifert fibered manifolds were introduced and investigated in [OW75]. Roughly speaking, these are (2n + 1)–manifolds L which admit a differentiable map f: L → X to a complex nmanifold X such that every fiber is a circle. Insisting that the base of a Seifert bundle be a complex manifold seems very