Distnace between points and isometric mapping

[paweld]

We define isometric mapping so that its tangent mapping preserves
the scalar product of vectors from tangent space (the definition
doesn't refer explicite to notion of distance in the manifold).
Distance between two points of manifold is the length of geodesics
which joins them.
I wonder if it's true that isometric mapping preserves distances
on the manifold?

 

[atyy]

It does. That's why integrated proper time is a "real thing" along a worldline which we can use to define as what an ideal clock should read.

 

[paweld]

Thanks, I thought so but I wasn't sure.

 

[Fredrik]

Keep in mind that the distance between points isn't well-defined in general. Instead we talk about the "proper time" of a timelike curve, and the "proper length" of a spacelike curve.

 

[paweld]

We might encounter even worse problems - sometimes there are points on the manifold
which cannot be connected by any geodesic or can be connected by more then
one geodesic. In such cases the notion of distance between these points isn't well defined.