Consider a Riemannian manifold $(M, g)$ which has some arbitrary connection specified on it. In general, such a connection could have both curvature and torsion, and the connection need not be compatible with the specified Riemmanian metric $g$ either.
My question is: Is there a "canonical" way to decompose such a connection into simpler pieces? For instance (this might be naive), would it be sensible to seek a decomposition into a sum of three terms which are Levi-Civita (i.e. torsion-free and metric compatible), metric-compatible with torsion, and a third piece which is neither?
Would such a decomposition be unique if it exists? I know the Levi-Civita connection is unique.
Thanks in advance.