On the conformal transformation of the sphere admitting an invariant connection (Q5935917)

Let \(M\) be a \(C^\infty\)-manifold and \(f: M\to M\) a \(C^\infty\)-diffeomorphism. A linear connection \(\nabla\) (maybe with non-zero torsion) on \(M\) transforms by \(f\) into a connection \(\nabla^f\), and it is said to be invariant with respect to \(f\) if \(\nabla^f = \nabla\). A Riemannian metric \(G\) on \(M\) transforms into a metric \(G^f\). Here \(f\) is said to be conformal if \(G^f = \varphi G\) for a \(C^\infty\)-function \(\varphi\) on \(M\), and invariant if \(G^f = G\). It is proved that a conformal transformation \(f\) of a standard sphere \(S^n\), \(n\neq 1,3,7\) (i.e., non-parallelizable) admits an invariant linear connection if and only if this \(f\) admits an invariant Riemannian metric. If \(n\) is even, the conformal transformations of \(S^n\), not admitting any invariant connection, constitute an open and dense subset in the group of all conformal transformations.