In the mathematical field of differential geometry, the fundamental theorem of surface theory deals with the problem of prescribing the geometric data of a submanifold of Euclidean space.
If these are smooth and satisfy the Gauss–Codazzi equations, then Bonnet's theorem says that D is covered by open sets which can be smoothly embedded into R3 with first fundamental form g and second fundamental form (relative to one of the two choices of unit normal vector field) h. Furthermore, each of these embeddings is uniquely determined up to a rigid motion of R3.
[2] Bonnet's theorem can be naturally formulated for hypersurfaces in a Euclidean space of any dimension, and the result remains true in this context.
This is more complicated to formulate, because in addition to the first and second fundamental forms, there is also the (generally nontrivial) connection in the normal bundle which must be taken into account.
[5] In this general context, the ambient Euclidean space can also be replaced by any connected and geodesically complete Riemannian manifold of constant curvature, which (as with the more special case of higher codimension) requires a suitably extended formulation of the Gauss–Codazzi equations.