# Distnace between points and isometric mapping

1. May 6, 2010

### 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?

2. May 6, 2010

### 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.

3. May 7, 2010

### paweld

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

4. May 7, 2010

### Fredrik

Staff Emeritus
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.

5. May 7, 2010

### 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.