In riemannin geometry, we define distance function by minimizing the length of curves. However we have nondefinite metric on psedu-riemannian manifold, so we cannot define a length of a curve as riemannian manifold $$L(\gamma)=\int_a^b<\dot\gamma,\dot\gamma>^{\frac{1}{2}}dt$$ Then we also cannot define the distance function, and complete concept on pseudo-riemannian manifold.
So how we define a complete pseudo-riemannian manifold? Can we just substitute the length with $$L(\gamma)=\int_a^b|<\dot\gamma,\dot\gamma>|^{\frac{1}{2}}dt$$
Any advice is helpful. Thank you.
This can presumably be done without the need for defining geodesic lengths. Observe that the notions of connections and parallel transport are perfectly valid for pseudo-Riemannian manifolds, and therefore we have the notion of the exponential map $\exp:TM \to M$ given by $\exp(p,v) = \exp_p(v) = \gamma(p,1,v)$ where $\gamma$ is the unique geodesic starting at point $p$ with initial tangent vector $v$, and the geodesic evaluated at "time" $1$. We can therefore define completeness of a pseudo-Riemannian manifold to mean that $\exp_p$ can be evaluated for all $v \in T_pM$. Note that this is equivalent to geodesics being extended to all of $\mathbb{R}$ as the real parameter.