An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures

Aloff, Simon; Wallach, Nolan R.

doi:10.1090/s0002-9904-1975-13649-4

Cited by 221 publications

(341 citation statements)

References 3 publications

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“…Furthermore, Perelman [103] developed a refined rescaling argument (by considering local limits and weak limits in Alexandrov spaces) for singularities of the Ricci flow on three-manifolds to obtain a uniform and global version of the canonical neighborhood structure theorem. We would like to point out that our proof of the singularity structure theorem (Theorem 7.1.1) is different from that of Perelman in two aspects: (1) we avoid using his crucial estimate in Claim 2 in Section 12.1 of [103]; (2) we give a new approach to extend the limit backward in time to an ancient solution. These differences are due to the difficulties in understanding Perelman's arguments at these points.…”

Section: References 486mentioning

confidence: 99%

“…For any two solutionsĝ (1) ij (·, t) andĝ (2) ij (·, t) of the Ricci flow (1.1.5) with the same initial data, we can solve the initial value problem (1.2.6) (or equivalently, (1.2.4)) to get two families ϕ (1) t and ϕ (2) t of diffeomorphisms of M . Thus we get two solutions, g (1) ij (·, t) = (ϕ (1) t ) * ĝ (1) ij (·, t) and g (2) ij (·, t) = (ϕ (2) t ) * ĝ (2) ij (·, t), to the modified evolution equation (1.2.5) with the same initial metric. The uniqueness result for the strictly parabolic equation (1.2.5) implies that g (1) ij = g (2) ij .…”

Section: Denote Bymentioning

confidence: 99%

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A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow

Cao¹,

Zhu²

2006

Asian Journal of Mathematics

303

365

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Section: References 486mentioning

confidence: 99%

Section: Denote Bymentioning

confidence: 99%

A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow

Cao¹,

Zhu²

2006

Asian Journal of Mathematics

303

365

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“…The non-normal homogeneous spaces discovered in [1] have the form (G,h)/K for this choice ofh, as described in [4].…”

Section: Metric Variationsmentioning

confidence: 99%

Obstruction to Positive Curvature on Homogeneous Bundles

Tapp

2006

Geom Dedicata

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Abstract. Examples of almost-positively and quasi-positively curved spaces of the form M = H\((G, h) × F ) were discovered recently [9], [8]. Here h is a left-invariant metric on a compact Lie group G, F is a compact Riemannian manifold on which the subgroup H ⊂ G acts isometrically on the left, and M is the orbit space of the diagonal left action of H on (G, h) × F with the induced Riemannian submersion metric. We prove that no new examples of strictly positive sectional curvature exist in this class of metrics. This result generalizes the case F = {point} proven by Geroch [5].

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“…This fact is very useful to systematically construct special holonomy manifolds with conical singularities, because the Einstein homogeneous spaces X m−1 = G/H endowed with these geometrical structures are well understood since the old days of Kaluza-Klein supergravity (SUGRA) [17][18][19][20][21][22][23] (and [24] for a review) as well as from the mathematical literature mentioned above [11][12][13][14][15][16] and [25].…”

Section: Introductionmentioning

confidence: 99%

Supercoset CFTs for string theories on non-compact special holonomy manifolds

Sugawara

Yamaguchi

2003

Nuclear Physics B

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We study aspects of superstring vacua of non-compact special holonomy manifolds with conical singularities constructed systematically using soluble N = 1 superconformal field theories (SCFT's). It is known that Einstein homogeneous spaces G/H generate Ricci flat manifolds with special holonomies on their cones ≃ R + × G/H, when they are endowed with appropriate geometrical structures, namely, the Sasaki-Einstein, triSasakian, nearly Kähler, and weak G 2 structures for SU(n), Sp(n), G 2 , and Spin (7) holonomies, respectively. Motivated by this fact, we consider the string vacua of the type:where we use the affine Lie algebras of G and H in order to capture the geometry associated to an Einstein homogeneous space G/H. Remarkably, we find the same number of spacetime and worldsheet SUSY's in our "CFT cone" construction as expected from the analysis of geometrical cones over G/H in many examples. We also present an analysis on the possible Liouville potential terms (cosmological constant type operators) which provide the marginal deformations resolving the conical singularities.

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An infinite family of distinct 7-manifolds admitting positively curved Riemannian structures

Cited by 221 publications

References 3 publications

A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow

A Complete Proof of the Poincaré and Geometrization Conjectures - application of the Hamilton-Perelman theory of the Ricci flow

Obstruction to Positive Curvature on Homogeneous Bundles

Supercoset CFTs for string theories on non-compact special holonomy manifolds

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