metric preserving transformations (isometry) Maybe it will be very general question but i wonder what is the importance of metric preserving transformations? Where can we use this concept in mathematics?
 A: In Euclidean geometry, rotations, translations and reflections are the basic types of isometries that have obvious importance and utility in real-life. They all preserve the notion of Euclidean distance and angle, which is what we're used to dealing with in everyday life.
In all likelihood, humans first realized that distance and angle were important, and then asked about what transformations left these notions alone and how they might describe them.
Changing metrics and exploring their isometry groups leads to all sorts of other geometries. In fact, many of those have physical relevance too, but not usually as obviously as the Euclidean example. For example, unitary geometry deals with transformations which preserve a Hermitian form on the space, and these are relevant for quantum phenomena.

Where can we use this in mathematics?

140 Years ago, there was a dude named Felix who said that the entirety of geometry should be studied this way. People listened. Grab a geometry book!
A: Sometimes the metric is easier to calculate in one system than another.  So transforming a problem from A to B and calculating the metric in B is the shortest path to a solution.
One example I've used is an area-preserving transformation from a sphere to a plane, to translate a polygon in spherical coordinates to planar and then calculating the area in the planar space.
