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
and
for all vector fields ,
,
and
on
.