Please excuse the unclarity in the question title but I don't know how to call the object I am looking for yet. My idea is that a metric on some topological space introduces a notion of lengths and angles and that there should be some other mathematical object that introduces a notion of angle only. (Such an object could among other things be of use to describe theories that are invariant under conformal transformations like electrodynamics.)
I thought about defining an object that is the equivalence class of some metric under conformal transformations but that seems quite involved. Is there a more basic notion?
Thank you very much.
If you are sufficiently comfortable with fiber bundles, you can read a very abstract definition of a conformal structure, for instance, in this book:
S.Kobayashi, "Transformation Groups in Differential Geometry".
In short, a conformal structure is a reduction of the structure group of the frame bundle of a smooth $n$-manifold $M$ from $GL(n,{\mathbb R})$ to the "conformal group" $CO(n):={\mathbb R}_+\cdot O(n)$.
For comparison, a Riemannian metric on $M$ can be defined as a reduction of the structure group from $GL(n,{\mathbb R})$ to $O(n)$.
The two notions are related through the fact that $CO(n)/O(n)\cong {\mathbb R}_+$, which is contractible. In concrete terms, this means that every conformal structure on $M$ can be defined as a conformal equivalence class of Riemannian metrics on $M$.