Compact transitive isometry spaces

Lukesh, Gordon W.; Kosniowski, Czes

doi:10.1017/cbo9781139106726.025

Cited by 2 publications

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“…Hence g is a U (i)-invariant metric on S 8 *" 1 and therefore the full isometry group of g is either U (k) or O(2k); see Lukesh [9]. In particular, Isom (g) cz O (2k) = Isom (i>).…”

Section: -1 Two-point Homogeneous Metricsmentioning

confidence: 99%

“…In subcase (i), M = S" or IRP" and the metric is two-point homogeneous and maximal symmetry follows from 4 and hence U (k) cz C(Z m ) cz Isom (g). Hence g is a U (i)-invariant metric on S 8 *" 1 and therefore the full isometry group of g is either U (k) or O(2k); see Lukesh [9]. In particular, Isom (g) cz O (2k) = Isom (i>).…”

Section: -2 Homogeneous Constant Curvature Riemannian Metrics Not Ementioning

confidence: 99%

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Maximally symmetric homogeneous metrics on manifolds

Bowers

1990

Math. Proc. Camb. Phil. Soc.

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IntroductionA metric d on a set has maximal symmetry provided its isometry group is not properly contained in the isometry group of any metric equivalent to d. This concept was introduced by Janos[7] and subsequently Williamson and Janos [17] proved that the standard euclidean metric on IR" has maximal symmetry. In Bowers [2], an elementary proof that every convex, complete, two-point homogeneous metric for which small spheres are connected has maximal symmetry is presented. This result in turn implies that the standard metrics on the classical spaces of geometryhyperbolic, euclidean, spherical and elliptic -are maximally symmetric. In this paper we study homogeneous metrics that possess maximal symmetry and, in particular, address the problem of the existence of such metrics and, to a lesser extent, their uniqueness.In Section 2, we use some well-known classical results to show that if the isometry group of a metric d o n a locally compact, connected, locally connected finitedimensional space X acts transitively on X, then X is a topological manifold and the isometry group Isom (d) can be given the structure of a Lie group. Consequently X admits a unique differentiable structure making Isom (d) into a Lie transformation group of X under its action by d-isometries and X admits a riemannian metric g whose isometry group contains Isom (d). The problem of finding a metric of maximal symmetry on X whose isometry group contains Isom (d) is reduced to the problem of finding a riemannian metric whose isometry group is maximal among the isometry groups of all riemannian metrics on X whose isometry groups contain Isom (d). The reduction of Section 2 takes our problem from the metric category and places it in the riemannian category.In Section 3, we show that an 'increasing sequence' of transitive groups of diffeomorphisms with compact point stabilizers of a fixed smooth manifold has an 'upper bound'. Coupled with the results of Section 2, this provides as an easy corollary that metrics of maximal symmetry exist on manifolds that admit a homogeneous riemannian metric, and that such maximally symmetric metrics can be chosen among those metrics induced by riemannian metrics. We close Section 3 with a discussion about the existence of metrics of maximal symmetry in the case that the manifold does not admit a metric with transitive isometry group.In Section 4, we present some examples of maximally symmetric metrics.In Section 5, we briefly address the problem of determining to what extent a homogeneous maximally symmetric metric is unique.Terminology and notation. The adjectives differentiable and smooth mean C x . A metric d o n a set is a distance function and a riemannian metric g on a smooth manifold is a smooth covariant tensor field of degree 2 that is positive definite and

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“…Hence g is a U (i)-invariant metric on S 8 *" 1 and therefore the full isometry group of g is either U (k) or O(2k); see Lukesh [9]. In particular, Isom (g) cz O (2k) = Isom (i>).…”

Section: -1 Two-point Homogeneous Metricsmentioning

confidence: 99%

Section: -2 Homogeneous Constant Curvature Riemannian Metrics Not Ementioning

confidence: 99%

Maximally symmetric homogeneous metrics on manifolds

Bowers

1990

Math. Proc. Camb. Phil. Soc.

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“…Then t(,)o ~ + (,)~ is also U(n -l)-invariant. In [5] and [6] we study such variations of metrics, and we are able to show that there are uncountably many homothetically distinct homogeneous metrics on the sphere S m= U(n)/U(n -1) having U(n) as full isometry group.…”

Section: Classification Theorem Let M = K/h Be An M-dimensional Compmentioning

confidence: 99%

Compact homogeneous Riemannian manifolds

Lukesh

1978

Geom Dedicata

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Compact transitive isometry spaces

Cited by 2 publications

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Maximally symmetric homogeneous metrics on manifolds

Maximally symmetric homogeneous metrics on manifolds

Compact homogeneous Riemannian manifolds

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