Geodesics on strong Kropina manifolds

Abstract

We study the behavior of the geodesics of strong Kropina spaces. The global and local aspects of geodesics theory are discussed. Our theory is illustrated with several examples.

Appendix

1.1 Conditions for a Kropina metric to be projectively equivalent to the Riemannian metric $\alpha$

Let be a Kropina space, where and . We denote by the symbol the covariant derivative with respect to Levi-Civita connection on the Riemannian space $(M, \alpha )$. The following notations are customary:

The geodesic spray coefficients of a Kropina space are expressed by

where

and are the coefficients of Levi-Civita connection with respect to $\alpha$.

We will get the conditions for the Kropina metric F to be projectively equivalent to the Riemannian metric $\alpha$. Suppose that the Kropina metric is projectively equivalent to Riemannian metric $\alpha$. Then there exists a function P on , which is positively homogeneous of degree one with respect to y, such that

(33)

(see [2, 18]).

Transvecting (33) by $y_i$, we get

that is,

(34)

Transvecting (33) by $b_i$, we get

$$\begin{aligned} -\,\frac{\beta r_{00}+\alpha ^2 s_0}{b^2\alpha ^2}\,\beta -\frac{\alpha ^2 s_0}{2\beta }+\frac{\beta r_{00}+\alpha ^2 s_0}{2\beta }=P\beta , \end{aligned}$$

and substituting (34) in the last equation, we obtain

that is,

$$\begin{aligned} \alpha ^2(b^2 r_{00}- s_0\beta ) =\beta ^2 r_{00}. \end{aligned}$$

(35)

Since $\beta ^2$ is not divisible by $\alpha ^2$, it follows that $r_{00}$ must be divisible by $\alpha ^2$, that is, there exists a function c(x) of x alone such that

(36)

Substituting (36) to (35), we have

(37)

Since must be divisible by $\beta$ and $\alpha ^2$ is not divisible by $\beta$, it follows that $c(x)=0$. Substituting $c(x)=0$ to (36) and (37), we have

$$\begin{aligned} r_{ij}=0, \qquad s_i=0. \end{aligned}$$

(38)

Then from (34) it follows that $P=0$. Lastly, substituting $P=0$ and (38) to (33), we get , that is,

$$\begin{aligned} s_{ij}=0. \end{aligned}$$

(39)

From (38) and (39), we get $b_{i;j}=0,$

that is, $b_i$ is parallel with respect to $\alpha$.

Conversely, suppose that $b_i$ is parallel with respect to $\alpha$, then we have , that is the Kropina metric is projectively equivalent to the Riemannian metric $\alpha$. Furthermore, we have .

Summarizing the above discussion, we obtain

Proposition 9.1

The necessary and sufficient condition for the Kropina metric to be projectively equivalent to the Riemannian metric $\alpha$ is that the vector field is parallel with respect to $\alpha$.

1.2 The condition $b_{i;j}=0$ in terms of the navigation data

By a straightforward computation we have

(40)

where , $k_i={\partial k}/{\partial x^i}$, and we put

where the notation “” stands for the covariant derivative with respect to h.

Suppose that the equation holds, then

$$\begin{aligned} r_{ij}= s_{ij}=0. \end{aligned}$$

(41)

Furthermore, from the equation it follows that $b^2$ is constant, hence the equation $b^2e^\kappa =4$ implies that $\kappa$ is also constant. Since , equations (40) reduce to

Hence we get

$$\begin{aligned} W_{i|j}=0. \end{aligned}$$

(42)

Conversely, suppose that equation (42) holds and that $\kappa$ is constant, then equation (40) reduces to (41), and therefore