Question

What is the a connection between isometry groups and Lie manifolds?

Answer 1

The isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation.

Its identity element is the identity function.The elements of the isometry group are called motions of the space.

Lie groups are differential manifolds whose group operations are smooth.

It is a continous group whose elements are several real parameters and it lies on the concept of continous symmetries.

Connection between the two is

Isometry group is a lie group in case of Riemannian symmetric spaces .