Codazzi equations | Mainardi-Codazzi equations

Let x:U->R^3 be a regular patch, where U is an open subset of R^2. Then (de)/(dv) - (df)/(du) | = | e Γ_12^1 + f(Γ_12^2 - Γ_11^1) - g Γ_11^2 (df)/(dv) - (dg)/(du) | = | e Γ_22^1 + f(Γ_22^2 - Γ_12^1) - g Γ_12^2, where e, f, and g are coefficients of the second fundamental form and Γ_(i j)^k are Christoffel symbols of the second kind. Let x:U->R^3 be a principal patch.

Christoffel symbol of the second kind | first fundamental form | second fundamental form