Followed by a topic entitled “the Second Fundamental Form” for classical geometry, we are here to discuss the Second Fundamental Form for Riemannian manifold.
Let be an immersion. For each , the inner product on splits into the direct sum
where is the orthogonal complement of in . If , , we can write
We call the tangential component of and the normal component of . Such a splitting is clearly differentiable, in the sense that the mappings
of into are differentiable.
The Riemannian connection on will be denoted by . If and are local vector fields on , and , are local extensions to , define
It is easy to verify that this is the Riemannian connection relative to the metric induced on . We want to define the second fundamental form of the immersion . To do this, it is convenient to introduce beforehand the following definition. If , are local vector fields on ,
is a local vector field on normal to , does not depend on the extensions , . Indeed, if is another extension of , we have
which vanishes on because on . Using what was proved, it follows that if is another extension of
because on .
Therefore is well-defined. In what follows, let us denote by the differentiable vector fields on that are normal to .
Proposition. If , the mapping given by
is bilinear and symmetric.
Because is bilinear, we see by writing in a system of coordinates, that the value depends only on the values and .
Now we are in a position to define the second fundamental form. Let and . The mapping given by
is a symmetric bilinear form.
Definition. The quadratic form defined on by
is called the second fundamental form of at along the normal vector .
Sometimes the expression second fundamental form is also used to designate the mapping which at every point is a symmetric bilinear mapping, taking values in .
Observe that the bilinear is associated to a linear self-adjoint operator by
The following proposition expresses the linear operator associated to the second fundamental form in terms of the covariant derivative.
Proposition. Let , and . Let be a local extension of normal to . Then