Christoffel Symbols in Differential Geometry
Buildup
Let $\mathbf{x} : U \to \mathbb{R}^{3}$ represent a coordinate mapping. In differential geometry, the characteristics and properties of geometric objects are described through differentiation. Therefore, the derivatives of the coordinate fragments $\mathbf{x}$ 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 $T_{p}M$. Hence, 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 consider the second-order derivatives of the coordinate mapping, $\mathbf{x}_{ij} = \dfrac{\partial^{2} \mathbf{x}}{\partial u_{i} \partial u_{j}}$. Since this is a vector of $\mathbb{R}^{3}$, it can be represented as a linear combination of the bases of $\mathbb{R}^{3}$. However, we already know three vectors in $\mathbb{R}^{3}$ that are orthogonal 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 expressed as follows:
$$ \mathbf{x}_{ij} = a_{ij} \mathbf{n} + b^{1}_{ij} \mathbf{x}_{1} + b^{2}_{ij} \mathbf{x}_{2} $$
These coefficients $b_{ij}^{1}, b_{ij}^{2}$ are called Christoffel symbols. Now, let’s concretely determine these coefficients. According to the properties of the first fundamental form, the following holds:
$$ \begin{align*} && \left\langle \mathbf{x}_{ij}, \mathbf{x}_{l} \right\rangle &=\ b_{ij}^{1}\left\langle \mathbf{x}_{1}, \mathbf{x}_{l} \right\rangle + b_{ij}^{2}\left\langle \mathbf{x}_{2}, \mathbf{x}_{l} \right\rangle \\ && &=\ \sum\limits_{k^{\prime}=1}^{2}b_{ij}^{k^{\prime}}\left\langle \mathbf{x}_{k^{\prime}}, \mathbf{x}_{l} \right\rangle \\ && &=\ \sum\limits_{k^{\prime}=1}^{2}b_{ij}^{k^{\prime}} g_{k^{\prime}l} \\ \implies && \sum\limits_{l=1}^{2}\left\langle \mathbf{x}_{ij}, \mathbf{x}_{l} \right\rangle g^{lk} &=\ \sum\limits_{l=1}^{2}\sum\limits_{k^{\prime}=1}^{2}b_{ij}^{k^{\prime}} g_{k^{\prime}l}g^{lk} \\ && &=\ \sum\limits_{k^{\prime}=1}^{2}b_{ij}^{k^{\prime}} \delta_{k^{\prime}}^{k} \\ && &=\ b_{ij}^{k} \end{align*} $$
Now let’s denote these $b_{ij}^{k}$ as $\Gamma_{ij}^{k}$ and define it as follows:
Definition
The $\Gamma_{ij}^{k}(1\le i,j,k \le 2)$ defined as follows is called the Christoffel symbol.
$$ \Gamma_{ij}^{k} := \sum \limits_{l=1}^{2} \left\langle \mathbf{x}_{ij}, \mathbf{x}_{l} \right\rangle g^{lk} = \left\langle \mathbf{x}_{ij}, \mathbf{x}_{l} \right\rangle g^{lk} $$
The omitted equation for $\sum$ uses the Einstein notation.
Explanation
Because of $\mathbf{x}_{12} = \mathbf{x}_{21}$, it follows that $\Gamma_{12}^{k} = \Gamma_{21}^{k}$.
The tangent components $b_{ij}^{k}$ of $\mathbf{x}_{ij}$ are denoted as $\Gamma_{ij}^{k}$ and called Christoffel symbols, and the normal component $a_{ij}$ of $\mathbf{x}_{ij}$ is denoted as $L_{ij}$ and called the coefficient of the second fundamental form.
The Christoffel symbols introduced above specifically refer to the second Christoffel symbol. The first Christoffel symbol is defined as follows.
$$ \Gamma_{ij \vert l} := \sum \limits_{k=1}^{2} \Gamma_{ij}^{k}g_{kl} $$
Usually, when referring to Christoffel symbols, it means the second symbol. G. B. Christoffel was the first to use these symbols, and at that time, the second symbol was written as $\begin{Bmatrix} ij \\ k \end{Bmatrix}$.