A general affine connection decomposes into metric, symmetric, mixed, and vector parts; in symmetric spacetimes the transverse-traceless part carries no local degrees of freedom and reduces to a residual gauge.
Rosenfeld, A history of non-Euclidean geom- etry : evolution of the concept of a geometric space (Springer-Verlag, 1988)
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Decomposition of the connection in affine models of gravity: Can the connection tell us something about the metric?
A general affine connection decomposes into metric, symmetric, mixed, and vector parts; in symmetric spacetimes the transverse-traceless part carries no local degrees of freedom and reduces to a residual gauge.