THE SCHWARZIAN DERIVAT IVE, CONFORMAL CONNECTIONS, AND MOBIUS STRUCTURES 8y

Despite its well-established role in complex analysis, the Schwarzian derivative remains omewhat mysterious. It is not a derivative, exactly, but what is it? The paper [0S1] takes up that question, introducing a generalized Schwarzian tensor S(f) when f is a conformal mapping between Riemannian manifolds. Upon closer examination, S(f) seems to derive not so much from the Riemannian metrics as from secondary geometric structures that the metrics determine. The present paper defines and studies those "Mrbius structures " which occupy a middle ground between Riemannian and conformal structures. Our main objective is to place the Schwarzian tensor in a natural geometric setting. That setting involves conformal connections structures that provide a notion of rolling the model sphere S ' ~ along a curve in a conformal manifold M. The conformal connections in M form an aft-me space in which the difference between elements ~r and H is naturally represented by a covariant ensor field A(~, H) of order two. More generally, if f is a conformal transformation from (M, H) to (M', H'), then H ' pulis back to a conformal connection f*H ' in M; and the tensor field A(f*H', H) describes the failure of f to intertwine H with H'. We call that tensor field,5(f) and declare f to be a Mrbius transformation i fS ( f) vanishes. Certain conformal connections are particularly compatible with Riemannian metrics; they will be called M6bius connections or M6bius structures. Every Riemannian metric determines such a connection. However, the relation is neither one-to-one nor onto: different metrics can have the same Mrbius connection, and not every M6bius connection arises from a Pdemannian metric. (See Theorem 5.3.) The generalized Schwarzian tensor is truest o its classical antecedent when applied to conformal transformations between manifolds with M6bius structures, in which case it is symmetric and trace-free. When those structures are the M6biu