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Reidegeld, Frank

doi:10.1016/j.geomphys.2010.03.006

Cited by 6 publications

(10 citation statements)

References 25 publications

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“…Cohomogeneity one actions are of independent interest in the field of group actions since they are the simplest examples of inhomogeneous actions. They also arise in physics as new examples of Einstein and EinsteinSasaki manifolds [Conti 2007, Gauntlett et al 2004] and as manifolds with G 2 and Spin(7)-holonomy [Cleyton and Swann 2002, Cvetič et al 2004, Reidegeld 2009. It is then interesting to ask how big the class of cohomogeneity one manifolds is.…”

Section: Introductionmentioning

confidence: 99%

Classification of cohomogeneity one manifolds in low dimensions

Hoelscher¹

2010

Pacific J. Math.

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A cohomogeneity one manifold is a manifold whose quotient by the action of a compact Lie group is one-dimensional. Such manifolds are of interest in Riemannian geometry in the context of nonnegative sectional curvature, as well as in other areas of geometry and in physics. We classify compact simply connected cohomogeneity one manifolds in dimensions 5, 6 and 7. We also show that all such manifolds admit metrics of nonnegative sectional curvature, with the possible exception of two families of manifolds.

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Section: Introductionmentioning

confidence: 99%

Classification of cohomogeneity one manifolds in low dimensions

Hoelscher¹

2010

Pacific J. Math.

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“…In order to decide if the holonomy is all of Spin (7), we need the following lemma. Lemma 3.15 (see [31]). (i) Let M be an eight-dimensional manifold that carries a parallel SU(4)-structure G. We denote the space of all parallel Spin (7)-structures on M which are an extension of G and have the same extension to an SO(8)-structure as G by S. Any connected component of S is diffeomorphic to a circle.…”

Section: Cohomogeneity-1 Manifoldsmentioning

confidence: 99%

“…We therefore have to check the smoothness conditions for the above power series. In [31], we have proved that an analytic diagonal metric g = g t + dt 2 of cohomogeneity 1 has a smooth extension to a singular orbit at t = 0 if (1) g t converges for t → 0 to a degenerate bilinear form that is invariant with respect to the cohomogeneity-1 action; (2) the sectional curvature of the collapsing sphere behaves as 1/t + O(1) for t → 0;…”

Section: (72)mentioning

confidence: 99%

Exceptional holonomy and Einstein metrics constructed from Aloff-Wallach spaces

Reidegeld

2011

Proceedings of the London Mathematical Society

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We investigate cohomogeneity-1 metrics whose principal orbit is an Aloff-Wallach space SU(3)/U (1). In particular, we are interested in metrics whose holonomy is contained in Spin (7). Complete metrics of this kind which are not product metrics have exactly one singular orbit. We prove classification results for metrics on tubular neighbourhoods of various singular orbits. Since the equation for the holonomy reduction has only few explicit solutions, we make use of power series techniques. In order to prove the convergence and the smoothness near the singular orbit, we apply methods developed by Eschenburg and Wang. As a by-product of these methods, we find many new examples of Einstein metrics of cohomogeneity 1.

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“…The complete examples produced by Bryant & Salamon have many symmetries, in fact the symmetry group acts with cohomogeneity one, so the principal orbit is of codimension one. Further systematic study of cohomogeneity one examples with compact symmetry group has been made by Reidegeld [34,35]. One sees that many of the examples and candidates have compact symmetry groups of rank 3, so an interesting class of Spin(7)-manifolds are those with T 3 -symmetry.…”

Section: Holonomy Spin(7)mentioning

confidence: 99%

Closed forms and multi-moment maps

Madsen

Swann

2012

Geom Dedicata

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We extend the notion of multi-moment map to geometries defined by closed forms of arbitrary degree. We give fundamental existence and uniqueness results and discuss a number of essential examples, including geometries related to special holonomy. For forms of degree four, multi-moment maps are guaranteed to exist and are unique when the symmetry group is (3,4)-trivial, meaning that the group is connected and the third and fourth Lie algebra Betti numbers vanish. We give a structural description of some classes of (3,4)-trivial algebras and provide a number of examples.Comment: 36 page

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Special cohomogeneity-one metrics withQ1,1,1orM1<

Cited by 6 publications

References 25 publications

Classification of cohomogeneity one manifolds in low dimensions

Classification of cohomogeneity one manifolds in low dimensions

Exceptional holonomy and Einstein metrics constructed from Aloff-Wallach spaces

Closed forms and multi-moment maps

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