The fundamental theorem of surface theory states that sufficiently smooth first fundamental form and second
fundamental form data on a simply connected
domain, with the first fundamental form positive
definite, determine an immersion into 
if they satisfy the Gauss-Codazzi
equations. The immersion is unique up to a rigid
motion. Thus, the Gauss-Codazzi equations are not only necessary compatibility equations for the two fundamental
forms, but also sufficient ones (Ciarlet 2003, Mardare
2004).
Fundamental Theorem of Surface Theory
See also
Compatibility Equations, First Fundamental Form, Gauss-Codazzi Equations, Second Fundamental FormExplore with Wolfram|Alpha
References
Ciarlet, P. G. "The Continuity of a Surface as a Function of Its Two Fundamental Forms." J. Math. Pures Appl. 82, 253-274, 2003. https://doi.org/10.1016/S0021-7824(03)00017-5.Mardare, S. "On the Fundamental Theorem of Surface Theory under Weak Regularity Assumptions." C. R. Acad. Sci. Paris 338, 71-76, 2004. https://doi.org/10.1016/j.crma.2003.10.027.Cite this as:
Weisstein, Eric W. "Fundamental Theorem of Surface Theory." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/FundamentalTheoremofSurfaceTheory.html