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

and

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 byis 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 byis 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.

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