On Finsler surfaces with certain flag curvatures

Taha, Ebtsam H.

doi:10.1142/s0219887823500020

Cited by 2 publications

(1 citation statement)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Further, ϕ is considered as a constraint which is a natural consequence of the Euler-Lagrange equations of motion. In addition, from the geometric point of view, the first integrals of the geodesic flow provide a lot of information about the underlying space such as rigidity results cf [27]. • Under the anisotropic conformal transformation F = e ϕ F, for two projectively related conic pseudo-Finsler surfaces, the projective flatness property is preserved.…”

Section: Discussionmentioning

confidence: 99%

Anisotropic conformal change of conic pseudo-Finsler surfaces, I^*

Youssef,

Elgendi,

Kotb

et al. 2024

Class. Quantum Grav.

Self Cite

View full text Add to dashboard Cite

The present work is devoted to investigate anisotropic conformal transformation of conic pseudo-Finsler surfaces $(M,F)$, that is, $ F(x,y)\longmapsto \overline{F}(x,y)=e^{\phi(x,y)}F(x,y)$, where the function $\phi(x,y)$ depends on both position $x$ and direction $y$, contrary to the ordinary (isotropic) conformal transformation which depends on position only. If $F$ is a pseudo-Finsler metric, the above transformation does not yield necessarily a pseudo-Finsler metric. Consequently, we find out necessary and sufficient condition for a (conic) pseudo-Finsler surface $(M,F)$ to be transformed to a (conic) pseudo-Finsler surface $(M,\overline{F})$ under the transformation $\overline{F}=e^{\phi(x,y)}F$. In general dimension, it is extremely difficult to find the anisotropic conformal change of the inverse metric tensor in a tensorial form. However, by using the modified Berwald frame on a Finsler surface, we obtain the change of the components of the inverse metric tensor in a tensorial form. This progress enables us to study the transformation of the Finslerian geometric objects and the geometric properties associated with the transformed Finsler function $\overline{F}$. In contrast to isotropic conformal transformation, we have a non-homothetic conformal factor $\phi(x,y)$ that preserves the geodesic spray. Also, we find out some invariant geometric objects under the anisotropic conformal change. Furthermore, we investigate a sufficient condition for $\overline{F}$ to be dually flat or/and projectively flat. Finally, we study some special cases of the conformal factor $\phi(x,y)$. Various examples are provided whenever the situation needs.

show abstract

Section: Discussionmentioning

confidence: 99%

Anisotropic conformal change of conic pseudo-Finsler surfaces, I^*

Youssef,

Elgendi,

Kotb

et al. 2024

Class. Quantum Grav.

Self Cite

View full text Add to dashboard Cite

show abstract

The geometry of geodesic invariant functions and applications to Landsberg surfaces

Elgendi,

Muzsnay

2024

MATH

View full text Add to dashboard Cite

<p>In this paper, for a given spray $ S $ on an $ n $-dimensional manifold $ M $, we investigated the geometry of $ S $-invariant functions. For an $ S $-invariant function $ {\mathcal P} $, we associated a vertical subdistribution $ {{\mathcal V}}_{\mathcal P} $ and found the relation between the holonomy distribution and $ {{\mathcal V}}_{\mathcal P} $ by showing that the vertical part of the holonomy distribution is the intersection of all spaces $ {{\mathcal V}}_{ {\mathcal F}_S} $ associated with $ {\mathcal F}_S $ where $ {\mathcal F}_S $ is the set of all Finsler functions that have the geodesic spray $ S $. As an application, we studied the Landsberg Finsler surfaces. We proved that a Landsberg surface with $ S $-invariant flag curvature is Riemannian or has a vanishing flag curvature. We showed that for Landsberg surfaces with non-vanishing flag curvature, the flag curvature is $ S $-invariant if and only if it is constant; in this case, the surface is Riemannian. Finally, for a Berwald surface, we proved that the flag curvature is $ H $-invariant if and only if it is constant.</p>

show abstract

On Finsler surfaces with certain flag curvatures

Cited by 2 publications

References 6 publications

Anisotropic conformal change of conic pseudo-Finsler surfaces, I^*

Anisotropic conformal change of conic pseudo-Finsler surfaces, I^*

The geometry of geodesic invariant functions and applications to Landsberg surfaces

Contact Info

Product

Resources

About

On Finsler surfaces with certain flag curvatures

Cited by 2 publications

References 6 publications

Anisotropic conformal change of conic pseudo-Finsler surfaces, I*

Anisotropic conformal change of conic pseudo-Finsler surfaces, I*

The geometry of geodesic invariant functions and applications to Landsberg surfaces

Contact Info

Product

Resources

About

Anisotropic conformal change of conic pseudo-Finsler surfaces, I^*

Anisotropic conformal change of conic pseudo-Finsler surfaces, I^*