Ngô Quốc Anh

January 6, 2010

R-G: The Second Fundamental Form, 2

Filed under: Riemannian geometry — Ngô Quốc Anh @ 1:55

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 f : M^n \to \overline M^{n+m=k} be an immersion. For each p \in M, the inner product on T_p\overline M splits T_p\overline M into the direct sum

\displaystyle T_p\overline M = T_pM \oplus (T_pM)^\bot,

where (T_pM)^\bot is the orthogonal complement of T_pM in T_p \overline M. If v \in T_p\overline M, p \in M, we can write

\displaystyle v=v^T +v^N, \quad v^T \in T_pM, v^N \in (T_pM)^\bot.

We call v^T the tangential component of v and v^N the normal component of v. Such a splitting is clearly differentiable, in the sense that the mappings

(p,v) \mapsto (p,v^T) and (p,v) \mapsto (p,v^N)

of T \overline M into T \overline M are differentiable.

The Riemannian connection on \overline M will be denoted by \overline \nabla. If X and Y are local vector fields on M, and \overline X, \overline Y are local extensions to \overline M, define

\displaystyle \nabla_XY=(\overline\nabla_{\overline X}\overline Y)^T.

It is easy to verify that this is the Riemannian connection relative to the metric induced on M. We want to define the second fundamental form of the immersion f : M \to \overline M. To do this, it is convenient to introduce beforehand the following definition. If X, Y are local vector fields on M,

\displaystyle B(X,Y)=\overline\nabla_{\overline X}\overline Y-\nabla_XY

is a local vector field on \overline M normal to M, B(X,Y) does not depend on the extensions \overline X, \overline Y. Indeed, if \overline X_1 is another extension of X, we have

\displaystyle (\overline\nabla_{\overline X}\overline Y-\nabla_XY)-(\overline\nabla_{\overline X_1}\overline Y-\nabla_XY)=\overline\nabla_{\overline X-\overline X_1}\overline Y,

which vanishes on M because \overline X - \overline X_1=0 on M. Using what was proved, it follows that if \overline Y_1 is another extension of Y

\displaystyle (\overline\nabla_{\overline X}\overline Y-\nabla_XY)-(\overline\nabla_{\overline X}\overline Y_1-\nabla_XY)=\overline\nabla_{\overline X}(\overline Y-\overline Y_1),

because \overline Y-\overline Y_1=0 on M.

Therefore B(X,Y) is well-defined. In what follows, let us denote by X(U)^\bot the differentiable vector fields on U that are normal to f(U).

Proposition. If X,Y \in X(U), the mapping B : X(U) \times X(U) \to X(U)^\bot given by

\displaystyle B(X,Y)=\overline\nabla_{\overline X}\overline Y-\nabla_XY

is bilinear and symmetric.

Because B is bilinear, we see by writing B in a system of coordinates, that the value B(X,Y)(p) depends only on the values X(p) and Y(p).

Now we are in a position to define the second fundamental form. Let p \in M and \eta \in (T_pM)^\bot. The mapping H_\eta : T_pM \times T_pM \to \mathbb R given by

\displaystyle {H_\eta }(x,y) = \left\langle {B(x,y),\eta } \right\rangle , \quad x,y \in {T_p}M,

is a symmetric bilinear form.

Definition. The quadratic form \mathrm I\!\mathrm I_\eta defined on T_pM by

\displaystyle\mathrm I\!\mathrm I_\eta(x) =H_\eta(x,x)

is called the second fundamental form of f at p along the normal vector \eta.

Sometimes the expression second fundamental form is also used to designate the mapping B which at every point p \in M is a symmetric bilinear mapping, taking values in (T_pM)^\bot.

Observe that the bilinear H_\eta is associated to a linear self-adjoint operator S_\eta : T_pM \to T_pM by

\displaystyle\left\langle {{S_\eta }(x),y} \right\rangle= {H_\eta }(x,y) = \left\langle {B(x,y),\eta } \right\rangle.

The following proposition expresses the linear operator associated to the second fundamental form in terms of the covariant derivative.

Proposition. Let p \in M, x\in T_pM and \eta \in (T_pM)^\bot. Let N be a local extension of \eta normal to M. Then

\displaystyle S_\eta(x)=-(\overline \nabla_xN)^T.

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