Second Fundamental Form in Differential Geometry
Build-up
Let $\mathbf{x} : U \to \mathbb{R}^{3}$ be referred to as a chart. In differential geometry, the characteristics and properties of geometric objects are explained through differentiation. Hence, the derivatives of coordinate charts $\mathbf{x}$ appear in various theorems and formulas. For instance, the first-order derivatives $\left\{ \mathbf{x}_{1}, \mathbf{x}_{2} \right\}$ become the basis of the tangent space $T_{p}M$. Therefore, any tangent vector $\mathbf{X} \in T_{p}M$ can be expressed as follows.
$$ \mathbf{X} = X^{1}\mathbf{x}_{1} + X^{2}\mathbf{x}_{2} $$
Now, let’s think about the second-order derivatives $\mathbf{x}_{ij} = \dfrac{\partial^{2} \mathbf{x}}{\partial u_{i} \partial u_{j}}$ of the chart. Since this is a vector in $\mathbb{R}^{3}$, it can be represented as a linear combination of the basis of $\mathbb{R}^{3}$. But we already know three vectors in $\mathbb{R}^{3}$ that are perpendicular to each other, which are the first-order derivatives and the unit normal.
$$ \left\{ \mathbf{n}, \mathbf{x}_{1}, \mathbf{x}_{2} \right\} $$
Then, $\mathbf{x}_{ij}$ can be represented as follows.
$$ \mathbf{x}_{ij} = a_{ij} \mathbf{n} + b^{1}_{ij} \mathbf{x}_{1} + b^{2}_{ij} \mathbf{x}_{2} $$
The coefficients $a_{ij} = \left\langle \mathbf{x}_{ij}, \mathbf{n} \right\rangle$ of the $\mathbf{n}$ terms of $\mathbf{x}_{ij}$ are called the coefficients of the second fundamental form of $\mathbf{x}$.
Definition
The inner product of $\mathbf{x}_{ij}$ and the unit normal $\mathbf{n}$, denoted as $L_{ij}$, is called the coefficients of the second fundamental form.
$$ L_{ij} := \left\langle \mathbf{x}_{ij}, \mathbf{n} \right\rangle $$
Let $\mathbf{X}, \mathbf{Y}$ be a vector in the tangent space $T_{P}M$ of the surface $\mathbf{x}$. Since the basis of the tangent space is $\left\{ \mathbf{x}_{1}, \mathbf{x}_{2} \right\}$, it can be represented as follows.
$$ \mathbf{X} = X^{1}\mathbf{x}_{1} + X^{2}\mathbf{x}_{2} \quad \text{and} \quad \mathbf{Y} = Y^{1}\mathbf{x}_{1} + Y^{2}\mathbf{x}_{2} $$
The following bilinear form $II$ is defined as the second fundamental form of the surface $\mathbf{x}$.
$$ II (\mathbf{X}, \mathbf{Y}) = \sum \limits_{i=1}^{2} \sum \limits_{j=1}^{2} L_{ij}X^{i}Y^{j} = L_{ij}X^{i}Y^{j} = \begin{bmatrix} X^{1} & X^{2}\end{bmatrix} \begin{bmatrix} L_{11} & L_{12} \\ L_{21} & L_{22} \end{bmatrix} \begin{bmatrix} Y^{1} \\ Y^{2}\end{bmatrix} $$
The omission of $\sum$ uses Einstein notation.
Explanation
Since $\mathbf{x}_{12} = \mathbf{x}_{21}$, then $L_{12} = L_{21}$.
The normal component $a_{ij}$ of $\mathbf{x}_{ij}$ is denoted as $L_{ij}$ and called the coefficient of the second fundamental form, and the tangential components $b_{ij}^{k}$ of $\mathbf{x}_{ij}$ are denoted as $\Gamma_{ij}^{k}$ and called Christoffel symbols.
If the first fundamental form was related to the function of the length of curves on the surface, then the second fundamental form is an indicator of how much the surface is curved, related to the Gaussian curvature $\kappa_{n}$ and related.
The first fundamental form is also called Riemannian metric, while the second fundamental form is simply referred to as the second fundamental form.