Abstract
On a non-degenerate hypersurface it is well known how to induce an affine connection ∇ and a symmetric bilinear form, called the affine metric. Conversely, given a manifold M and an affine connection ∇ one can ask whether this connection is locally realizable as the induced affine connection on a nondegenerate affine hypersurface and to what extend this immersion is unique. In case that the image of the curvature tensor R of ∇ is 2-dimensional and M is at least 3-dimensional a rigidity theorem was obtained in [4]. In this paper, we discuss positive definite n-dimensional affine hypersurfaces with rank 1 shape operator (which is equivalent with 1-dimensional image of the curvature tensor) which are non-rigid. We show how to construct such affine hypersurfaces using solutions of (n - 1)-dimensional differential equations of Monge–Ampère type.
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