On compositions of conformal immersions

Abstract:

We consider conformal immersions of a manifold ${M^n},\;n \geqslant 6$, into conformally flat manifolds. If the principal curvatures of $f:{M^n} \to N_{cf}^{n + 1}$ have multiplicities at most $n - 4$, we show that any $g:{M^n} \to \tilde N_{cf}^{n + 2}$ can locally be written as $g = \rho \circ f$, where $\rho :N_{cf}^{n + 1} \to \tilde N_{cf}^{n + 2}$ is a conformal immersion.