On homogeneous manifolds of negative curvature

Heintze, Ernst

doi:10.1007/bf01344139

Cited by 197 publications

(139 citation statements)

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“…Let N = [S, S], and let S, N be the Lie algebras of S and N . If X is a unit length vector in S which is orthogonal to N, then S = RX ⊕ N and (up to changing X to −X) the eigenvalues of the derivation A = ad X | N of N have positive real parts (see [Hei,Prop. 2] and [EH,Sect.…”

Section: Unclouding the Sky Of Negatively Curved Manifoldsmentioning

confidence: 99%

Unclouding the sky of negatively curved manifolds

Parkkonen

Paulin²

2005

GAFA, Geom. funct. anal.

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Abstract. Let M be a complete simply connected Riemannian manifold, with sectional curvature K ≤ −1. Under some assumptions on the geometry of ∂M , which are satisfied for instance if M is a symmetric space, or has dimension 2, we prove that given any family of horoballs in M , and any point x0 outside these horoballs, it is possible to shrink uniformly, by a finite amount depending only on M , these horoballs so that some geodesic ray starting from x0 avoids the shrunk horoballs. As an application, we give a uniform upper bound on the infimum of the heights of the closed geodesics in the finite volume quotients of M .

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Section: Unclouding the Sky Of Negatively Curved Manifoldsmentioning

confidence: 99%

Unclouding the sky of negatively curved manifolds

Parkkonen

Paulin²

2005

GAFA, Geom. funct. anal.

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“…Results of J. A. Wolf [13] and E. Heintze [4] show that the above problem is closely linked with the classification of connected, homogeneous Riemannian manifolds with negative curvature. Indeed, if M is such a manifold and if M is simply connected, then M is isometric to a solvable Lie group endowed with a left-invariant metric.…”

mentioning

confidence: 99%

“…For the case of strictly negative curvature (see §7.8), the problem was recently solved by E. Heintze [4]. Here one looks only at those AC algebras where the derived subalgebra has codimension 1 and most of the technical difficulties inherent in the general situation are not present.…”

mentioning

confidence: 99%

Homogeneous manifolds with negative curvature. I

Azencott¹,

Wilson²

1976

Trans. Amer. Math. Soc.

103

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ABSTRACT. This paper solves the problem of determining which Lie groups act simply transitively on a Riemannian manifold with negative curvature. The results obtained extend those of Heintze for the case of strictly negative curvature.Using results of Wolf and Heintze, it is established that every connected, simply connected, homogeneous manifold M with negative curvature admits a Lie group S acting simply transitively by isometries and every group with this property must be solvable. Formulas for the curvature tensor on M are established and used to show that the Lie algebra of any such group S must satisfy a number of structural conditions. Conversely, given a Lie algebra < satisfying these conditions and any member of an easily constructed family of inner products on i, a metric deformation argument is used to obtain a modified inner product which gives rise to a left invariant Riemannian structure with negative curvature on the associated simply connected Lie group.1. Introduction. This paper was motivated by the following problem: Which connected Lie groups admit a left invariant Riemannian metric with negative (sectional) curvature? We emphasize that throughout the paper, we understand "negative" to mean "less than or equal to zero". Since the property in question is not sensitive to groups linked by a local isomorphism, we deal primarily with simply connected groups. Results of J. A. Wolf [13] and E. Heintze [4] show that the above problem is closely linked with the classification of connected, homogeneous Riemannian manifolds with negative curvature. Indeed, if M is such a manifold and if M is simply connected, then M is isometric to a solvable Lie group endowed with a left-invariant metric.In this paper, we give a complete solution to our original problem by showing that a necessary and sufficient condition for a group to have the property in question is that its Lie algebra be what we call an "NC algebra". Roughly speaking, the crucial properties of an NC algebra $ are that in addition to being solvable, é must contain an abelian subalgebra a complementary to the derived

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“…Indeed, it was in the 1970's that the structure of homogeneous Riemannian manifolds of nonpositive curvature was determined. More precisely, in 1974, Heintze [4] proved that a connected, simply connected homogeneous Riemannian manifold of nonpositive curvature can be identified with a simply connected solvable Lie group with a left invariant metric. In consequence, to classify the structure of these manifolds it suffices to determine the structure of solvable Lie algebras g with inner product , of nonpositive curvature.…”

Section: Introductionmentioning

confidence: 99%

“…In this direction, Heintze [4] studied a necessary and sufficient condition for a metric solvable Lie algebra (g, , ) to have strictly negative sectional curvature, and obtained the condition that (g, , ) be isomorphic to the metric Lie algebra associated with a Riemannian symmetric space of negative curvature. Subsequently, in 1976, Azencott and Wilson [1] succeeded in determining the structure of metric solvable Lie algebras (g, , ) of nonpositive curvature.…”

Section: Introductionmentioning

confidence: 99%

Homogeneous Kähler Einstein manifolds of nonpositive curvature operator

Obata¹

2007

Tohoku Math. Publ.

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On homogeneous manifolds of negative curvature

Cited by 197 publications

References 7 publications

Unclouding the sky of negatively curved manifolds

Unclouding the sky of negatively curved manifolds

Homogeneous manifolds with negative curvature. I

Homogeneous Kähler Einstein manifolds of nonpositive curvature operator

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