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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.

4 Did a lot of revision of the question.

But say I am working on a pseudo-Riemannian manifold with $(1,n)$ signature. Call a curve to be time-like" if the tangent vector to it is always of negative norm andspace-like" if it is always positive and 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 metric 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, but I am not exactly clear about the claim.

Confusingly

Confusion begins when I also see being 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 distance length function on the space of rectifiable curves between between two fixed points becomes what is called an "upper semi-continuous" function on in the space of rectifiable paths between those two pointsabove 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 on it such that the norm of its tangent vector is between them.

In light of the same parity all along the curve. () theorem I guess this existence of a length maximizing geodesic the above only says that between any two points with constant parity tangent vector norm is very special to on a globally hyperbolic manifolds unlike the existence of space-time there exists a length minimizing geodesic between any two with no conjugate points which is needed for the D-function to make sense.

Is one assuming that the D-function in between and then by the Frechet Toplogy continue to make sense even on a pseudo-Riemannian manifolds?

More confusingly in () statement it will automatically be the above topology "proper local maximum length " of the curves is apparently NOT a continuous function on the space of rectifiable curves. I don't have a clue as to what is this referring totime-like curve between them.

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