Help with this proof: if K is a timelike killing vector field, what does the notations $g(K, K)$ and $g(K, \gamma^{\prime})$ denote?

Question

I need some help with this proof. In particular, I don’t understand the notations $g(K, K)$ and $g(K, \gamma^{\prime})$, what they denote exactly?

Moreover, why $\inf\vert g(K, K)\vert >0$ implies that it is enough to check that $g(K, \gamma^{\prime})$ is bounded?

Thank you in advance

Answer 1

We want to show that the projection of $\gamma'$ to the line bundle $L_K$ defined by $K$ is bounded. The fibre of $L_K$ at $x$ is $L_K=\{cK(x),c\in\mathbb{R}\}$. We can take a Riemannian metric and rescale $K$ (multiply $K$ by ${1\over{\|K(x)\|}}$) such that $\|K(x)\|=1$ for this metric.

Let $U_K$ the orthogonal to $L_K(x)$, we have $\gamma'(x)=f(x)K(x)+v(x)$ where $v(x)\in U_K$ and the $p(\gamma'(x))$ the projection of $\gamma'(x)$ is $f(x)K(x)$. We have $g(K(x),\gamma'(x))=g(f(x)K(x)+v(x),K(x))=f(x)g(K(x),K(x))$.

Suppose that $g(K(x),\gamma'(x))<M$, it implies that $|f(x)|<{M\over{|g(K(x),K(x)))|}}<{M\over C}$. We deduce that $\|p(\gamma'(x))\|<|f(x)|\|K(x)\|<{M\over C}$. Where $C=Inf(K(x),K(x))_{x\in M}$.