On Locally Dually Flat Finsler Metrics

Cheng, Xinyue; Zhang, Shen; Zhou, Yusheng

doi:10.1142/s0129167x10006616

Cited by 34 publications

(19 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Later on, Shen extended the notion of locally dually flatness for Finsler metrics [47]. He identified and studied the dually flat Finsler metrics that are a special and valuable class of Finsler metrics in Finsler geometry, which play a very important role in studying flat Finsler information structures [16].…”

Section: Information Geometrymentioning

confidence: 99%

A five-dimensional Riemannian manifold with an irreducible SO(3)-structure as a model of abstract statistical manifold

Mikeš

Stepanova

2013

Ann Glob Anal Geom

View full text Add to dashboard Cite

In the present paper, we consider a five-dimensional Riemannian manifold with an irreducible SO(3)-structure as an example of an abstract statistical manifold. We prove that if a five-dimensional Riemannian manifold with an irreducible SO(3)-structure is a statistical manifold of constant curvature, then the metric of the Riemannian manifold is an Einstein metric. In addition, we show that a five-dimensional Euclidean sphere with an irreducible SO(3)-structure cannot be a conjugate symmetric statistical manifold. Finally, we show some results for a five-dimensional Riemannian manifold with a nearly integrable SO(3)-structure. For example, we prove that the structure tensor of a nearly integrable SO(3)-structure on a five-dimensional Riemannian manifold is a harmonic symmetric tensor and it defines the first integral of third order of the equations of geodesics. Moreover, we consider some topological properties of five-dimensional compact and conformally flat Riemannian manifolds with irreducible SO(3)-structure.

show abstract

Section: Information Geometrymentioning

confidence: 99%

A five-dimensional Riemannian manifold with an irreducible SO(3)-structure as a model of abstract statistical manifold

Mikeš

Stepanova

2013

Ann Glob Anal Geom

View full text Add to dashboard Cite

show abstract

“…(ᾱ,β) is called the navigation data of the Randers metric F . Based on the results in [4], the author provided a more clear description for dually flat Randers metrics: A Randers metric F = α + β is locally dually flat if and only if α is locally dually flat andβ is dually related with respect toᾱ [16]. The notion of dually related 1-forms was proposed by the author in [16].…”

Section: )mentioning

confidence: 99%

“…are dually flat Randers metrics, where α and β are given by (1.7), µ and σ are constant numbers. When µ = −1 and σ = 1, the corresponding metrics are the famous generalized Funk metrics, whose dual flatness were first proved in [4] The Finsler metric…”

Section: )mentioning

confidence: 99%

On dually flat general (α,β)-metrics

2015

Differential Geometry and its Applications

View full text Add to dashboard Cite

Based on the previous researches, in this paper we study the dual flatness of a special class of Finsler metrics called general (α, β)-metrics, which is defined by a Riemannian metric α and a 1-form β. A characterization for such metrics to be locally dually flat under some suitable conditions is provided. Many non-trivial explicit examples are constructed by a new kind of deformation technique. Moreover, the relationship of dual flatness and project flatness of such metrics are shown.

show abstract

“…Such a coordinate system is called an adapted coordinate system [2,7,9,10]. In [5], Shen proved that F on an open subset U ⊂ R n is dually flat if and only if it satisfies…”

Section: Introductionmentioning

confidence: 99%