I am looking for references on the relationship between the curvature of a manifold and the representation theory of some group related to the manifold.
I've once heard that the curvature tensor, Ricci, etc, appear in some way as irreducible representations.
Thanks in advance.
The Holonomy Groups are very important.
A connection is a way of defining a derivative on a manifold. Once your have a connection, you can talk about how something, e.g. a tangent vector, changes as you move it around your manifold.
If you parallel transport something, e.g. a tangent vector, around a manifold then you move it in a way that the connection says it hasn't changed.
The interesting thing is that you can parallel transport something around a closed loop, but have something different at the end than what you started with! The connection says it didn't change, but the things at the start and at the end are different.
If you moved tangent vectors, then the start vector and end vector can be related by a linear transformation. If you look at all possible tangent vectors, and all possible loops (with the same start and end point) then you get a set of linear transformations. These form the holonomy group of the connection.
