Search

In this paper, we study a class of Finsler metrics called general $$(\alpha ,\beta )$$ -metrics, which are defined by a Riemannian metric and a 1-form. These metrics form a rich class of Finsler metrics. Suppose that $$F=\alpha \phi (b^{2},s)$$ is a general $$(\alpha ,\beta )$$ -metric which is conformally related to $$\tilde{F}=e^{\sigma (x)}F,$$ where $$\sigma := \sigma (x)$$ is a scalar function on M. We show that if F is Douglas metric, then $$\tilde{F}$$ is also Douglas metric if and only if the conformal transformation is a homothety under suitable condition on $$\phi $$ . Furthermore, we have investigated the conformal transformation of isotropic S-curvature general $$(\alpha ,\beta )$$ -metrics.

In this paper, we study $$\Xi $$ -curvature and H-curvature of a special class of Finsler metrics called general $$(\alpha ,\beta )$$ -metrics. We prove that every general $$(\alpha ,\beta )$$ -metric of almost vanishing H-curvature is of almost vanishing $$ \Xi $$ -curvature under certain conditions. Moreover, we study such Finsler metric with vanishing $$\Xi $$ -curvature and its interaction to the flag curvature.

Finsler metrics of quadratic curvatures form a rich and important class of metrics. In this paper, we study spherically symmetric metrics of quadratic curvatures. We find equations that characterize R-quadratic spherically symmetric Finsler metrics. In addition, we find the necessary and sufficient conditions for the metrics to be W-quadratic and Ricci-quadratic types.

In this paper, we investigate the flag curvature of a special class of Finsler metrics called general spherically symmetric Finsler metrics, which are defined by a Euclidean metric and two related 1-forms. We find equations to characterize the class of metrics with constant Ricci curvature (tensor) and constant flag curvature. Moreover, we study general spherically symmetric Finsler metrics with the vanishing non-Riemannian quantity χ-curvature. In particular, we construct some new projectively flat Finsler metrics of constant flag curvature.

Projective Ricci curvature is a projective invariant quantity in Finsler geometry which is introduced by Z. Shen. In this paper, we study special projective Ricci curvature of C-reducible Finsler metrics. The necessary and sufficient conditions of these metrics, which cause these metrics to be weak or isotropic projective Ricci curvature, are found and it is proved that C-reducible Douglas metric of isotropic PRic-curvature must be PRic flat. The same theorem for C-reducible metrics of scalar flag curvature is also investigated.

Microinjection is a widely used technique to inject defined volumes and concentrations of substances and explore their physiological function in vivo. The technique has been particularly successful with zebrafish embryos; however, the injection equipment can be relatively expensive and therefore available only to well-funded laboratories. In this study, a simple, cheap, easy-to-assemble, and easy-to-use setup with a straightforward, accurate, and efficacious calibration method is introduced. The accuracy of this calibration method was tested by comparing with the results of calibration methods that are currently used in high-cost systems. Injection success with this low-cost system was verified based on the presence of injected dyes in zebrafish embryos, the absence of any significant morphological and behavioral differences between 3,4,-dichloroaniline-treated and untreated embryos, and larval viability.

This article derives some theoretical relations to predict the instantaneous axial force of the circular metal tubes during the splitting process under the axial compression by the theoretical and experimental methods. A new theoretical model of splitting deformation of the circular tube as a thin-walled structure subjected to the axial loading is introduced and based on the deformation model and using the energy method, some analytical formulas are derived to estimate the instantaneous axial load, maximum splitting load, steady force and curl radius of the tubes. To verify the present theory, some aluminum, mild steel and brazen tubes with the circular cross-section and different geometrical and material characteristics were prepared and axially compressed on conical dies with the different cone angles. Comparison of the theoretical predictions and the experimental measurements shows a good correlation and it affirms verity of the new theoretical deformation model and the present theory. Also, based on the experimental results, the effects of wall thickness and inner radius of the tubes and also, number and length of initial slits and semi-angle of conical die are studied on the axial load and maximum splitting load.

The local structure of Finsler metrics of constant flag curvature have been historically mysterious. It is proved that every Matsumoto metric of constant flag curvature on a closed n-dimensional manifold of dimension n ≥ 3 is either Riemannian or locally Minkowskian.

It is proved that a Randers metric F = α + β on a manifold of dimension n ⩾ 3 is projective if and only if the Lie algebra of projective vector fields p ( M , F ) has (locally) dimension n ( n + 2 ) . This can be regarded as an analogue of the corresponding result in Riemannian geometry.

The collection of all projective vector fields on a Finsler space $(M, F)$ is a finite-dimensional Lie algebra with respect to the usual Lie bracket, called the projective algebra denoted by $p(M,F)$ and is the Lie algebra of the projective group $P(M,F)$. The projective algebra $p(M,F=\alpha+\beta)$ of a Randers space is characterized as a certain Lie subalgebra of the projective algebra $p(M,\alpha)$. Certain subgroups of the projective group $P(M,F)$ and their invariants are studied. The projective algebra of Randers metrics of constant flag curvature is studied and it is proved that the dimension of the projective algebra of Randers metrics constant flag curvature on a compact $n$-manifold either equals $n(n+2)$ or at most is $\frac{n(n+1)}{2}$.

In this paper, we study projective algebra, p(M, F), of special (α, β)-metrics. The projective algebra of a Finsler space is a finite-dimensional Lie algebra with respect to the usual Lie bracket. We characterize p(M, F) of Matsumoto and square metrics of isotropic S-curvature of dimension n ≥ 3 as a certain Lie sub-algebra of the Killing algebra k(M, α). We also show that F has a maximum projective symmetry if and only if F either is a Riemannian metric of constant sectional curvature or locally Minkowskian.