Compatibility equations are conditions that separately prescribed data must satisfy in order to arise from a single mathematical object. For the first
fundamental form and second fundamental
form of a surface in 
, compatibility means that the two forms can arise together
from a single surface. Its intrinsic geometry and extrinsic bending cannot be prescribed
independently, and the relations enforcing this requirement are the Gauss-Codazzi
equations.
The Gauss-Codazzi equations are necessary for the fundamental forms of every regular surface.
Conversely, for sufficiently smooth form data
on a simply connected domain, with the first
fundamental form positive definite,
they are also sufficient for the existence of an immersion into 
. This converse is the fundamental
theorem of surface theory (Ciarlet 2003, Mardare 2004).
