A vector field $k^{a}$ which satisfies

$$k_{a ; b}+k_{b ; a}=0$$

is called a Killing vector field. Prove that $k^{a}$ is a Killing vector field if and only if

$$k^{c} g_{a b, c}+k_{, b}^{c} g_{a c}+k_{, a}^{c} g_{b c}=0$$

Prove also that if $V^{a}$ satisfies

$$V_{; b}^{a} V^{b}=0$$

then

$$\left(V^{a} k_{a}\right)_{, b} V^{b}=0$$

for any Killing vector field $k^{a}$.

In the two-dimensional space-time with coordinates $x^{a}=(u, v)$ and line element

$$d s^{2}=-d u^{2}+u^{2} d v^{2}$$

verify that $(0,1), e^{-v}\left(1, u^{-1}\right)$ and $e^{v}\left(-1, u^{-1}\right)$ are Killing vector fields. Show, by using $(*)$ with $V^{a}$ the tangent vector to a geodesic, that geodesics in this space-time are given by

$$\alpha e^{v}+\beta e^{-v}=2 \gamma u^{-1}$$

where $\alpha, \beta$ and $\gamma$ are arbitrary real constants.