Riemannian geometry studies smooth manifolds that are equipped with a Riemannian metric: Riemannian manifolds.
Riemannian geometry is hence equivalently the Cartan geometry for inclusions o the orthogonal group into the Euclidean group.
In index theory:
Isaac Chavel, Riemannian geometry – A modern introduction Cambridge University Press (1993)
Marcel Berger, A panoramic view of Riemannian geometry