# 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

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