Affine transformations in affine differential geometry (Q1826131)

For a \(C^{\infty}\) hypersurface immersion f: \(M^ n\to R^{n+1}\), with M orientable, let \(\nabla\) be the affine connection induced by the affine normal and let h be the corresponding (first affine) fundamental form. The author studies the Lie subgroup of those affine transformations of M, with respect to \(\nabla\), that preserve h, namely \(G=\{\phi \in A(M,\nabla):\quad \phi^*h=h\},\) and proves a rigidity result. Theorem: If \(\phi\in G\), then there exists a unique \({\tilde \phi}\) affinity of \(R^{n+1}\) with \({\tilde \phi}\circ f=f\circ \phi\), where f is the non degenerate immersion of M. Next, he finds a necessary and sufficient condition for an affine transformation in A(M,\(\nabla)\) to belong to G in the case where M is, in addition, compact. Within this context he further proves that if A(M,\(\nabla)\) acts transitively on M, then f is a diffeomorphism between M and an ellipsoid. Finally, a similar result to the last is proven for the case of a centro-affine immersion of a compact manifold.