Conformal immersions of warped products

Abstract

We prove a decomposition theorem for conformal immersions \(f\colon\;M^n\to {\mathbb{R}}^{N}\) into Euclidean space of a warped product of Riemannian manifolds \(M^n:=M_0\times_\rho\Pi_{i=1}^k M_i\) of dimension n ≥ 3 under the assumption that the second fundamental form \(\alpha \colon TM \times TM\to T^\perp M\) of f satisfies \(\alpha|_{TM_i\times TM_j}=0\) for i ≠ j. It generalizes the corresponding theorem of Nolker for isometric immersions as well as our previous result on conformal immersions of Riemannian products. In particular, we determine all conformal representations of Euclidean space of dimension n ≥ 3 as a warped product of Riemannian manifolds. As a consequence, we classify the conformally flat warped products.