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 Homogeneous Finsler spaces and the flag-wise positively curved condition

2018 ◽  
Vol 30 (6) ◽  
pp. 1521-1537
Author(s):  
Ming Xu ◽  
Shaoqiang Deng
Keyword(s):  
Lie Algebra ◽  
Tangent Plane ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Finsler Spaces ◽  
Negatively Curved ◽  
Hopf Conjecture ◽  
Coset Spaces ◽  
Positive Flag Curvature ◽  
Positively Curved

Abstract In this paper, we introduce the flag-wise positively curved condition for Finsler spaces (the (FP) condition), which means that in each tangent plane, there exists a flag pole in this plane such that the corresponding flag has positive flag curvature. Applying the Killing navigation technique, we find a list of compact coset spaces admitting non-negatively curved homogeneous Finsler metrics satisfying the (FP) condition. Using a crucial technique we developed previously, we prove that most of these coset spaces cannot be endowed with positively curved homogeneous Finsler metrics. We also prove that any Lie group whose Lie algebra is a rank 2 non-Abelian compact Lie algebra admits a left invariant Finsler metric satisfying the (FP) condition. As by-products, we find the first example of non-compact coset space {S^{3}\times\mathbb{R}} which admits homogeneous flag-wise positively curved Finsler metrics. Moreover, we find some non-negatively curved Finsler metrics on {S^{2}\times S^{3}} and {S^{6}\times S^{7}} which satisfy the (FP) condition, as well as some flag-wise positively curved Finsler metrics on {S^{3}\times S^{3}} , shedding some light on the long standing general Hopf conjecture.

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.

scholarly journals The Β-Change by Finsler Metric of C-Reducible Finsler Spaces in Finsler Geometry

2019 ◽  
Vol 33 (1) ◽  
pp. 1-10
Author(s):  
Khageswar Mandal
Keyword(s):  
Finsler Space ◽  
Homogeneous Function ◽  
Finsler Geometry ◽  
Finsler Metric ◽  
Finsler Spaces

 This paper considered about the β-Change of Finsler metric L given by L*= f(L, β), where f is any positively homogeneous function of degree one in L and β and obtained the β-Change by Finsler metric of C-reducible Finsler spaces. Also further obtained the condition that a C-reducible Finsler space is transformed to a C-reducible Finsler space by a β-change of Finsler metric.

scholarly journals A NEW QUANTITY IN RIEMANN-FINSLER GEOMETRY

2012 ◽  
Vol 54 (3) ◽  
pp. 637-645 ◽  
Author(s):  
XIAOHUAN MO ◽  
ZHONGMIN SHEN ◽  
HUAIFU LIU
Keyword(s):  
Scalar Curvature ◽  
Simple Proof ◽  
Finsler Geometry ◽  
Finsler Manifold ◽  
Flag Curvature ◽  
Finsler Metric ◽  
Finsler Metrics ◽  
Riemannian Curvature

AbstractIn this note, we study a new Finslerian quantity Ĉ defined by the Riemannian curvature. We prove that the new Finslerian quantity is a non-Riemannian quantity for a Finsler manifold with dimension n = 3. Then we study Finsler metrics of scalar curvature. We find that the Ĉ-curvature is closely related to the flag curvature and the H-curvature. We show that Ĉ-curvature gives, a measure of the failure of a Finsler metric to be of weakly isotropic flag curvature. We also give a simple proof of the Najafi-Shen-Tayebi' theorem.

scholarly journals Multiple Closed Geodesics on Positively Curved Finsler Manifolds

2019 ◽  
Vol 19 (3) ◽  
pp. 495-518 ◽  
Author(s):  
Wei Wang
Keyword(s):  
Finsler Manifold ◽  
Flag Curvature ◽  
Closed Geodesics ◽  
Finsler Manifolds ◽  
Positively Curved

Abstract In this paper, we prove that on every Finsler manifold {(M,F)} with reversibility λ and flag curvature K satisfying {(\frac{\lambda}{\lambda+1})^{2}<K\leq 1} , there exist {[\frac{\dim M+1}{2}]} closed geodesics. If the number of closed geodesics is finite, then there exist {[\frac{\dim M}{2}]} non-hyperbolic closed geodesics. Moreover, there are three closed geodesics on {(M,F)} satisfying the above pinching condition when {\dim M=3} .

scholarly journals On complete Finsler spaces of constant negative Ricci curvature

2020 ◽  
Vol 17 (03) ◽  
pp. 2050041
Author(s):  
Behroz Bidabad ◽  
Maryam Sepasi
Keyword(s):  
Ricci Curvature ◽  
Schwarzian Derivative ◽  
Finsler Space ◽  
Ricci Scalar ◽  
Flag Curvature ◽  
Finsler Spaces ◽  
Randers Metric

Here, using the projectively invariant pseudo-distance and Schwarzian derivative, it is shown that every connected complete Finsler space of the constant negative Ricci scalar is reversible. In particular, every complete Randers metric of constant negative Ricci (or flag) curvature is Riemannian.

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)}}.

scholarly journals The singular terms in the fundamental matrix of crystal optics

2011 ◽  
Vol 467 (2133) ◽  
pp. 2663-2689 ◽  
Author(s):  
Peter Wagner
Keyword(s):  
Explicit Formula ◽  
Fundamental Matrix ◽  
Singular Part ◽  
Wave Surface ◽  
Crystal Optics ◽  
Cauchy Principal ◽  
Singular Terms ◽  
Negatively Curved ◽  
Principal Value ◽  
Positively Curved

We derive an explicit formula for the singular part of the fundamental matrix of crystal optics. It consists of a singularity remaining fixed at the origin x =0, of delta terms located on the positively curved parts of the wave surface, the well-known Fresnel surface and of a Cauchy principal value distribution on the negatively curved part of the wave surface.

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