Global variant of Bonnet theorem for Hilbert space. (English. Russian original) Zbl 0645.58006

Sov. Math. 31, No. 3, 96-100 (1987); translation from Izv. Vyssh. Uchebn. Zaved., Mat. 1987, No. 3(298), 71-73 (1987).
If M is a hypersurface of \(R^{n+1}\), then its second fundamental form A may be regarded as a tensor field of type (1,1) that defines a symmetric linear transformation of each tangent space \(T_ x(M)\) and satisfies the equations of Gauss and Codazzi. Conversely, if an n-dimensional Riemannian manifold M admits a symmetric tensor field A of type (1,1) that satisfies the equations of Gauss and Codazzi, then for each point x of M there exists a neighborhood \(U_ x\subseteq M\) of x and an essentially unique isometric imbedding \(f_ x: U_ x\to R^{n+1}\) such that A is the second fundamental form of the hypersurface \(f_ x(U_ x)\). Briefly, a Riemannian manifold is determined locally by its first and second fundamental forms. If \(n=2\) this result is due to Bonnet. In this paper the author announces a global analogue of this result for manifolds M modeled on a real separable Hilbert space, but the hypotheses require in addition the information of the third fundamental form of M.
Reviewer: P.Eberlein


58B20 Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds