On Flag Curvature of Homogeneous Randers Spaces

2013 ◽  
Vol 65 (1) ◽  
pp. 66-81 ◽  
Author(s):  
Shaoqiang Deng ◽  
Zhiguang Hu
Keyword(s):  
Explicit Formula ◽  
Lie Group ◽  
Flag Curvature ◽  
Nilpotent Lie Group ◽  
Finsler Spaces ◽  
Rigidity Result ◽  
Randers Space ◽  
Negatively Curved

AbstractIn this paper we give an explicit formula for the flag curvature of homogeneous Randers spaces of Douglas type and apply this formula to obtain some interesting results. We first deduce an explicit formula for the flag curvature of an arbitrary left invariant Randersmetric on a two-step nilpotent Lie group. Then we obtain a classification of negatively curved homogeneous Randers spaces of Douglas type. This results, in particular, in many examples of homogeneous non-Riemannian Finsler spaces with negative flag curvature. Finally, we prove a rigidity result that a homogeneous Randers space of Berwald type whose flag curvature is everywhere nonzero must be Riemannian.

Geodesic orbit Finsler spaces with K ≥ 0 and the (FP) condition

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Ming Xu
Keyword(s):  
Lie Algebra ◽  
Finsler Space ◽  
Finsler Manifold ◽  
Constant Speed ◽  
Flag Curvature ◽  
Finsler Metric ◽  
Coset Space ◽  
Finsler Spaces ◽  
Negatively Curved ◽  
Positively Curved

Abstract We study the interaction between the g.o. property and certain flag curvature conditions. A Finsler manifold is called g.o. if each constant speed geodesic is the orbit of a one-parameter subgroup. Besides the non-negatively curved condition, we also consider the condition (FP) for the flag curvature, i.e. in any flag we find a flag pole such that the flag curvature is positive. By our main theorem, if a g.o. Finsler space (M, F) has non-negative flag curvature and satisfies (FP), then M is compact. If M = G/H where G has a compact Lie algebra, then the rank inequality rk 𝔤 ≤ rk 𝔥+1 holds. As an application,we prove that any even-dimensional g.o. Finsler space which has non-negative flag curvature and satisfies (FP) is a smooth coset space admitting a positively curved homogeneous Riemannian or Finsler metric.

scholarly journals KILLING FRAMES AND S-CURVATURE OF HOMOGENEOUS FINSLER SPACES

2014 ◽  
Vol 57 (2) ◽  
pp. 457-464 ◽  
Author(s):  
MING XU ◽  
SHAOQIANG DENG
Keyword(s):  
Explicit Formula ◽  
Vector Fields ◽  
Finsler Space ◽  
Killing Vector ◽  
Finsler Spaces ◽  
Killing Vector Fields ◽  
Randers Space ◽  
Homogeneous Finsler Space ◽  
Special Case

AbstractIn this paper, we first deduce a formula of S-curvature of homogeneous Finsler spaces in terms of Killing vector fields. Then we prove that a homogeneous Finsler space has isotropic S-curvature if and only if it has vanishing S-curvature. In the special case that the homogeneous Finsler space is a Randers space, we give an explicit formula which coincides with the previous formula obtained by the second author using other methods.

scholarly journals On seven-dimensional quaternionic contact solvable Lie groups

2014 ◽  
Vol 26 (2) ◽  
Author(s):  
Diego Conti ◽  
Marisa Fernández ◽  
José A. Santisteban
Keyword(s):  
Heisenberg Group ◽  
Lie Groups ◽  
Lie Group ◽  
Contact Structure ◽  
Contact Manifold ◽  
Nilpotent Lie Group ◽  
Solvable Lie Groups ◽  
Quaternionic Heisenberg Group

AbstractWe answer in the affirmative a question posed by Ivanov and Vassilev on the existence of a seven-dimensional quaternionic contact manifold with closed fundamental 4-form and non-vanishing torsion endomorphism. Moreover, we show an approach to the classification of seven-dimensional solvable Lie groups having an integrable left invariant quaternionic contact structure. In particular, we prove that the unique seven-dimensional nilpotent Lie group admitting such a structure is the quaternionic Heisenberg group.

scholarly journals Reversible homogeneous Finsler metrics with positive flag curvature

2017 ◽  
Vol 29 (5) ◽  
pp. 1213-1226 ◽  
Author(s):  
Ming Xu ◽  
Wolfgang Ziller
Keyword(s):  
Flag Curvature ◽  
Finsler Metrics ◽  
Isotropy Group ◽  
Difficult Case ◽  
Finsler Spaces ◽  
Fixed Point Set ◽  
Point Set ◽  
Positive Flag Curvature ◽  
Positively Curved

AbstractIn this work, we continue with the classification for positively curve homogeneous Finsler spaces {(G/H,F)}. With the assumption that the homogeneous space {G/H} is odd dimensional and the positively curved metric F is reversible, we only need to consider the most difficult case left, i.e. when the isotropy group H is regular in G. Applying the fixed point set technique and the homogeneous flag curvature formulas, we show that the classification of odd dimensional positively curved reversible homogeneous Finsler spaces coincides with that of L. Bérard Bergery in Riemannian geometry except for five additional possible candidates, i.e. {\mathrm{SU}(4)/\mathrm{SU}(2)_{(1,2)}\mathrm{S}^{1}_{(1,1,1,-3)}}, {\mathrm{Sp}(2)/\mathrm{S}^{1}_{(1,1)}}, {\mathrm{Sp}(2)/\mathrm{S}^{1}_{(1,3)}}, {\mathrm{Sp}(3)/\mathrm{Sp}(1)_{(3)}\mathrm{S}^{1}_{(1,1,0)}}, and {G_{2}/\mathrm{SU}(2)} with {\mathrm{SU}(2)} the normal subgroup of {\mathrm{SO}(4)} corresponding to the long root. Applying this classification to homogeneous positively curved reversible {(\alpha,\beta)} metrics, the number of exceptional candidates can be reduced to only two, i.e. {\mathrm{Sp}(2)/\mathrm{S}^{1}_{(1,1)}} and {\mathrm{Sp}(3)/\mathrm{Sp}(1)_{(3)}\mathrm{S}^{1}_{(1,1,0)}}.

Lie group classification of the N-th-order nonlinear evolution equations

2011 ◽  
Vol 54 (12) ◽  
pp. 2553-2572 ◽  
Author(s):  
ShouFeng Shen ◽  
ChangZheng Qu ◽  
Qing Huang ◽  
YongYang Jin
Keyword(s):  
Lie Group ◽  
Evolution Equations ◽  
Nonlinear Evolution ◽  
Group Classification ◽  
Nonlinear Evolution Equations ◽  
Lie Group Classification
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