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 $\widetilde\nabla$ 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 $\nabla$ 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 $\nabla$ is related to the Levi-Civita connection $\nabla^{\ast}$ of $h$ by the formula
\[
\nabla_XY = \nabla_X^{\ast}Y + \varphi(X)Y + \varphi(Y)X + h(X,Y)E,
\]
$\varphi$ 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In this paper, certain ideas from my unpublished report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.