Riemannian Connection On A Surface - Gauss-Codazzi Equations

Gauss-Codazzi Equations

When M is embedded in E3, two other 1-forms ψ and χ can be defined on the frame bundle E using the shape operator. Indeed the Gauss map induces a K-equivariant map of E into SO(3), the frame bundle of S2 = SO(3)/SO(2). The form ω is the pullback of one of the three right invariant Maurer-Cartan forms on SO(3). The 1-forms ψ and χ are defined to be the pullbacks of the other two.

These 1-forms satisfy the following structure equations:

(symmetry equation)

(Gauss equation)

(Codazzi equations)

The Gauss–Codazzi equations for χ, ψ and ω follow immediately from the Maurer-Cartan equations for the three right invariant 1-forms on SO(3).