# Gaussian curvature and mean curvature sufficient to characterize a surface?

Is the knowledge of Gaussian and mean curvature (and thus of the principal curvatures) sufficient to characterize a surface uniquely?

If not, is there another geometric quantity one can add to obtain a unique characterization?

• It is interesting to compare your question with this one and the answers to it. Apr 17, 2014 at 4:36

The Bonnet theorem (see also here) states, roughly speaking, that if we are given the data $$\tilde{I}$$ and $$\tilde{II}$$ that are proposed to be the first and the second fundamental forms of the surface, then provided they satisfy the equations of Gauss and Codazzi (and Ricci in a more general setting) there exist a surface in the Euclidean space such that its first and second fundamental forms are $$I=\tilde{I}$$ and $$II=\tilde{II}$$ respectively. Such a surface is unique up to a rigid motion of the embracing space.
The principal curvatures $$\kappa_1,\,\kappa_2$$ are the eigenvalues of the shape operator $$I^{-1}II$$. Knowing them is equivalent to knowing the Gauss $$K=\kappa_1 \cdot \kappa_2$$ and the mean $$H=\tfrac{1}{2}\left( \kappa_1+\kappa_2 \right)$$ curvatures. This knowledge is insufficient because the problem would be underdetermined: given the eigenvalues of the matrix $$I^{-1}II$$, there is a lot of freedom to choose the matrices $$I$$ and $$II$$.