On naturally reductive left-invariant metrics of SL (2, R)

Halverscheid, S.; Iannuzzi, Andrea

doi:10.2422/2036-2145.2006.2.03

Cited by 9 publications

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“…For G = SL 2 (R) we give a precise description of these complexifications and we determine their basic complex-geometric properties. For positive m this gives, along with previous results (see [Sz2], [BHH], [HaIa2]), examples among all classes of 3-dimensional, naturally reductive, Riemannian homogeneous spaces (cf. [BTV]).…”

Section: Introductionsupporting

confidence: 79%

“…The Riemannian ones appear in the classification given by C. Gordon in [Go]. More details and curvature computations can be found in [HaIa2].…”

Section: Preliminariesmentioning

confidence: 99%

“…From [Go], section 5, it follows that in the Riemannian cases m > 0 the connected component of the isometry group is essentially given by G × K (here discrete ineffectivity is allowed) while in the pseudo-Riemannian symmetric case m = −1 it coincides with G × G. Further information regarding the Levi Civita connections (or their limit when m vanishes), the curvature tensor, the scalar and Ricci curvature can be found in [HaIa2], section 3.…”

Section: Preliminariesmentioning

confidence: 99%

“…However (U, ν) has some negative sectional curvatures (see [DZ], cf. [HaIa2], Sect. 3), thus by Theorem 2.4 in [LeSz] the adapted complex structure is not defined on all of T U.…”

Section: Preliminariesmentioning

confidence: 99%

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A family of adapted complexifications for SL2(R)

Halverscheid¹,

Iannuzzi²

2009

Annali Scuola Normale Superiore - Classe Di Scienze

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Let G be a non-compact, real semisimple Lie group. We consider maximal complexifications of G which are adapted to a distinguished oneparameter family of naturally reductive, left-invariant metrics. In the case of G = SL 2 (R) their realization as equivariant Riemann domains over G C = SL 2 (C) is carried out and their complex-geometric properties are investigated. One obtains new examples of non-univalent, non-Stein, maximal adapted complexifications.

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Section: Introductionsupporting

confidence: 79%

“…The Riemannian ones appear in the classification given by C. Gordon in [Go]. More details and curvature computations can be found in [HaIa2].…”

Section: Preliminariesmentioning

confidence: 99%

Section: Preliminariesmentioning

confidence: 99%

“…However (U, ν) has some negative sectional curvatures (see [DZ], cf. [HaIa2], Sect. 3), thus by Theorem 2.4 in [LeSz] the adapted complex structure is not defined on all of T U.…”

Section: Preliminariesmentioning

confidence: 99%

See 2 more Smart Citations

A family of adapted complexifications for SL2(R)

Halverscheid¹,

Iannuzzi²

2009

Annali Scuola Normale Superiore - Classe Di Scienze

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“…Topologically M 3 Γ = SM 2 Γ is the unit tangent bundle of the surface M 2 Γ and carries out a class of natural metrics, coming from the left-invariant metrics on SL(2, R), which are also right SO(2) invariant. They are particular case of the two-parameter family of the naturally reductive metrics [22] on SL(n, R), which are left SL(n, R)-invariant and right SO(n)-invariant and determined by the following inner product on the Lie algebra sl(n, R) :…”

Section: Introductionmentioning

confidence: 99%

Chaos and integrability in SL(2,R)-geometry

Bolsinov,

Veselov,

2019

Preprint

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The integrability of the geodesic flow on the three-folds M 3 admitting SL(2, R)-geometry in Thurston's sense is investigated.The main examples are the quotients M 3 Γ = Γ\P SL(2, R), where Γ ⊂ P SL(2, R) is a cofinite Fuchsian group. We show that the corresponding phase space T * M 3Γ contains two open regions with integrable and chaotic behaviour. In the integrable region we have Liouville integrability with analytic integrals, while in the chaotic region the system is not Liouville integrable even in smooth category and has positive topological entropy.As a concrete example we consider the case of modular 3-fold with the modular group Γ = P SL(2, Z), when M 3 Γ is known to be homeomorphic to the complement in 3-sphere of the trefoil knot K. Ghys proved a remarkable fact that the lifts of the periodic geodesics on the modular surface to M 3 Γ produce the same class of knots, which appeared in the chaotic version of the celebrated Lorenz system. We show that in the integrable limit of the geodesic system on M 3 Γ we have the class of the satellite knots known as the cable knots of trefoil.

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The classification of naturally reductive homogeneous spaces in dimensions n≤6

Agricola

Ferreira

Friedrich

2015

Differential Geometry and its Applications

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We present a new method for classifying naturally reductive homogeneous spaces -i. e. homogeneous Riemannian manifolds admitting a metric connection with skew torsion that has parallel torsion and curvature. This method is based on a deeper understanding of the holonomy algebra of connections with parallel skew torsion on Riemannian manifolds and the interplay of such a connection with the geometric structure on the given Riemannian manifold. It allows to reproduce by easier arguments the known classifications in dimensions 3, 4, and 5, and yields as a new result the classification in dimension 6. In each dimension, one obtains a 'hierarchy' of degeneracy for the torsion form, which we then treat case by case. For the completely degenerate cases, we obtain results that are independent of the dimension. In some situations, we are able to prove that any Riemannian manifold with parallel skew torsion has to be naturally reductive. We show that a 'generic' parallel torsion form defines a quasi-Sasakian structure in dimension 5 and an almost complex structure in dimension 6. within the priority programme 1388 "Representation theory". Ana Ferreira thanks Philipps-Universität Marburg for its hospitality during a research stay in May-July 2013, and she also acknowledges partial financial support by the FCT through the project PTDC/MAT/118682/2010 and the University of Minho through the FCT project PEst-C/MAT/UI0013/2011. We also thank Andrew Swann (Aarhus) for discussions on Section 4 during a research visit to Marburg in June 2013 and Simon G. Chiossi (Marburg) for many valuable comments on a preliminary version of this work. Metric connections with skew torsionConsider a Riemannian manifold (M n , g). The difference between its Levi-Civita connection ∇ g and any linear connection ∇ is a (2, 1)-tensor field A,The curvature of ∇ resp. ∇ g will always be denoted by R resp. R g . Following Cartan, we study the algebraic types of the torsion tensor for a metric connection. Denote by the same symbol the (3, 0)-tensor derived from a (2, 1)-tensor by contraction with the metric. We identify T M n with (T M n ) * using g from now on. Let T be the n 2 (n − 1)/2-dimensional space of all possible torsion tensors,A connection ∇ is metric if and only if A belongs to the spaceThe spaces T and A are isomorphic as O(n) representations, reflecting the fact that metric connections can be uniquely characterized by their torsion. For n ≥ 3, they split under the action of O(n) into the sum of three irreducible representations,

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On naturally reductive left-invariant metrics of SL (2, R)

Cited by 9 publications

References 0 publications

A family of adapted complexifications for SL2(R)

A family of adapted complexifications for SL2(R)

Chaos and integrability in SL(2,R)-geometry

The classification of naturally reductive homogeneous spaces in dimensions n≤6

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