Conformal and Isometric Immersions of Conformally Flat Riemannian Manifolds into Spheres and Euclidean Spaces

Abstract

We consider n-dimensional compact Riemannian manifolds (n ≧ 3) which are conformally flat if n ≧ 4 and give as well global topological (if n≧ 4) and metric obstructions for the existence of a conformai immersion into the N-dimensional sphere SN with N≦ 2n-2 (which are due to [Moore 2] and [Moore 3]) as local metric obstructions for the existence of an isometric immersion into S or Euclidean space EN . Then we apply these results to examples of conformally flat manifolds as space forms, products of space forms with opposite curvature and warped products of S1 and a nonspherical space form.