On hypersurfaces in Banach manifolds (Q1884082)

The author establishes Gauss-Codazzi equations for hypersurfaces in a Banach manifold and a generalization of the Schur theorem to the case of Banach manifolds. Concepts of bending and equi-affinity are introduced for infinite-dimensional hypersurfaces in Hilbert manifolds. The following theorems are proven: 1) A hypersurface \(N\) of constant sectional curvature in a Hilbert manifold \(M\) can be bent into a hyperplane if \(\beta= 0\), and into a hypersphere if \(\beta>0\); 2) The necessary and sufficient condition for a Hilbert manifold to be locally isomorphic to a manifold of constant sectional curvature is that it has the same constant curvature. Finally is proven: 3) The locally equi-affine hypersurfaces of a Hilbert manifold \(M\) of constant sectional curvature are the only hyperspheres.