Now at days the concept of a complete Riemannian manifold with bounded geometry is quite standard (see e.g. J. Cheeger, M. Gromov, M. Taylor, Finite propagation speed, kernel estimates for functions of the laplace operator, and the geometry of complete riemannian manifolds). It consists of two conditions. One one hand, one ask for positivity of the injectivity radius; and on the other hand, there are two equivalent conditions:
a) Boundedness of the metric tensor and its derivatives on any geodesic ball,
b) Boundedness of the curvature tensor and its covariant derivatives.
This equivalence works because one is working with the Levi-Civita connection of the manifold. So I am wondering if anyone has ever seen this definition for more general connections, in particular I am interested in the case when the connection is not necessarily metric (it could be torsion-free). I think a possible definition could be the restriction on the injectivity radius, boundedness of the metric tensor and its derivatives, and boundedness of the Christoffel symbols and its derivatives. But I am not sure if this is really useful or not.
Thanks for any information.