On nondegenerate umbilical affine hypersurfaces in recurrent affine manifolds

Abstract

Let $widetilde{M}$ be a differentiable manifold of dimension $geqslant 5$, which is endowed with a (torsion-free) affine connection $widetildeabla$ of recurrent curvature. Let $M$ be a nondegenerate umbilical affine hypersurface in $widetilde{M}$, whose shape operator does not vanish at every point of $M$. Denote by $abla$ and $h$, respectively, the affine connection and the affine metric induced on $M$ from the ambient manifold. Under the additional assumption that the induced connection $abla$ is related to the Levi-Civita connection $abla^{ast}$ of $h$ by the formula [ abla_XY = abla_X^{ast}Y + varphi(X)Y + varphi(Y)X + h(X,Y)E, ] $varphi$ being a $1$-form and $E$ a vector field on $M$, it is proved that the affine metric $h$ is conformally flat. Relations to totally umbilical pseudo-Riemannian hypersurfaces are also discussed. In this paper, certain ideas from my unpublished report [14] (cf. also [15]) are generalized.