general relativity - Is there a single metric for a given system?

Let imagine a tunnel that connect two distant places at the globe (eastern-western or north-south)

There are a lot of posible "distances" or metrics, defined by maps, routes, "as the crow flies", etc.. but none of those distance can be shorter than the distance of the tunel.

So if two trains travels at same speed, one inside the tunnel and other above in the surface, the one on the tunnel will reach first.

If this is possible, then perhaps it's possible to have differents coexisting metrics with differents dispositions or topologies, within the same system.

Of course that if we describe a space-time metric surrounding a sphere, then "holes" in it would change the metric (just because it's not a sphere anymore). But it's strange for me that making a hole we could change in some way the space-time shape.

In an extreme case. Could be an euclidian space of same dimension be build within a non-euclidian space?

I would like to have a view from people familiar with general theory of relativity, thanks