So far, we have discussed Codazzi equations in classical differential geometry. Today, we discuss the Gauss and Codazzi equations in Riemannian geometry. Later on, we will discuss in a new entry the Gauss and Codazzi equations in general relativity.
A classical problem in geometry is to determine whether a Riemannian manifold can be isometrically immersed in another Riemaniann manifold . We will restrict ourselves to the case of codimension immersions, i.e., has dimension and has dimension .
It is well known that the Gauss and Codazzi equations are necessary conditions relating the Riemann curvature tensor of , the Riemann curvature tensor of and the shape operator of (or the second fundamental form). Denoting by the Riemannian connection of , these equations are the following
for all vector fields , , and on .