Let $(M,g)$ be a Riemannian manifold and let $p$ and $q$ be two points on it and define $d(p,q)$ as the length of the minimizing geodesic between them. Now given two rectifiable paths $\gamma_1$ and $\gamma_2$ connecting $p$ and $q$ on $M$ and parametrized proportional to their length one defines a ``distance" between the paths $\gamma_1$ and $\gamma_2$, $D(\gamma_1,\gamma_2)$ as,
$D(\gamma_1,\gamma_2) = Sup_{\lambda \in [0,1]} d(\gamma_1 (\lambda), \gamma_2 (\lambda)) + \vert l_g(\gamma _1) - l_g (\gamma _2)\vert $
where $l_g(\gamma)$ is the length of the curve $\gamma$ in the metric $g$.
For the above definition is it necessary that $g$ be complete as is often assumed? Further when all is the existence of a length minimizing geodesic guaranteed between any two points? (I am aware of the result which says that given any point there is always a neighbourhood around it where between any two points there is a length minimizing geodesic)
But say I am working on a pseudo-Riemannian manifold with $(1,n)$ signature. Call a curve to be time-like"time-like" if the tangent vector to it is always of negative norm and space-like"space-like" if it is always positive and ``null" "null" if it is always zero.
(*)
Then I know of a theorem which proves that a smooth time-like curve connecting two points realizes the local maximum of length between these points if and only if it is a geodesic between them with no conjugate points in between.
How does this fit in with the need to have a unique locally minimizing geodesic for the D-function to be defined?
I think that the topology introduced by this distance function on the space of paths between $p$ and $q$ is what is called the ``Frechet Topology" and under which this space of paths becomes an infinite dimensional metric space and it is also complete. I think under this topology the distance function between two points on the manifold becomes a continuous function on the space of curves joining the two points.
Confusion begins when I see claims which seem to mean that in the $(1,n)$ signature case the "proper length" of time-like curves is apparently NOT a continuous function on the space of rectifiable curves in the above topology.
(Is one assuming that the above definition of the D-function and the Frechet Toplogy continue to make sense even on a pseudo-Riemannian manifolds?)
But it is claimed that the length function on the space of rectifiable curves between between two fixed points becomes what is called an "upper semi-continuous" function in the above topology. (I don't have much intuition about this.)
Apparently these kind of functions satisfy the good old property of attaining a maxima on compact sets and this gets used to say that on a globally hyperbolic pseudo-Riemannian manifold given any two points there is a length maximizing time-like geodesic between them.
In light of the () theorem stated in the fourth paragraph I guess the above only says that between any two points on a globally hyperbolic space-time there exists a geodesic with no conjugate points in between and then by the () that statement it will automatically be the local maximum length time-like curve between them.
I would like to know what is the intuition behind this definition of ``distance" on the path spaces and how the alleged consequences follow. Also if someone can give me the bigger picture of what is going on.
