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My colleague and friend, Mario Micheli, has co-authored (with P.Michor, D.Mumford) a paper “Sectional Curvature in terms of the Cometric, with Applications to the Riemannian Manifolds of Landmarks” (arxiv). This is a landmark (sic!) paper, that expresses the curvature of the manifold of landmarks in terms of the cometric, which is a much simpler formula (but by no means simple) than the expression in terms of the metric (cometric is the inverse of the metric).

Another interesting part is that authors decomposed the curvature into four terms, thus attempting to describe what exactly makes curvature positive/negative. The decomposition is given in physical terms (strain, force, compression). Armed with a strong physical intuition one might be able to decipher what exactly makes the curvature non-zero.

A more general paper on the expression of curvature in terms of cometric is coming out later, as I have been told by MM.

Lowering or raising indices or, in other words, canonical isomorphisms between a tangent bundle $TM$ and a cotangent bundle $T^*M$ are called musical isomorphisms.