On the Existence of Almost Affinely Flat Structure Induced by Hypersurface Immersion on Connected Compact Manifold

Abstract

Given a hypersurface immersion and a transversal vector field, the formula of Gauss leads to an induced connection and a symmetric bilinear function called affine fundamental form. We define the norm of tensor field using the affine fundamental form (assumed to be nondegenerate) and prove that a hypersurface immersion on a connected compact 𝑛-dimensional differential manifold 𝑀 into the affine space ℝ𝑛+1 induces an almost affinely flat structure on 𝑀.