Some flaws

It’s a pity that do Carmo didn’t add up any material arguing the consistency of notions (affine connections, in particular Levi-Civita connections, and Gauss curvature, etc.) in the general setting of Riemmanian manifold in arbitrary dimensions and those defined in terms of differential forms on Riem. manifolds, though one can roughly see some indication of them. However, even exercises aiming for showing these kind of consistencies are rare to be found.