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Christoffel Symbols in Differential Geometry 📂Geometry

Christoffel Symbols in Differential Geometry

Buildup

Let x:U→R3\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 x\mathbf{x} appear in various theorems and formulas. For instance, the first-order derivatives {x1,x2}\left\{ \mathbf{x}_{1}, \mathbf{x}_{2} \right\} become the basis of the tangent space TpMT_{p}M. Hence, any tangent vector X∈TpM\mathbf{X} \in T_{p}M can be expressed as follows:

X=X1x1+X2x2 \mathbf{X} = X^{1}\mathbf{x}_{1} + X^{2}\mathbf{x}_{2}

Now consider the second-order derivatives of the coordinate mapping, xij=∂2x∂ui∂uj\mathbf{x}_{ij} = \dfrac{\partial^{2} \mathbf{x}}{\partial u_{i} \partial u_{j}}. Since this is a vector of R3\mathbb{R}^{3}, it can be represented as a linear combination of the bases of R3\mathbb{R}^{3}. However, we already know three vectors in R3\mathbb{R}^{3} that are orthogonal to each other, which are the first-order derivatives and the unit normal.

{n,x1,x2} \left\{ \mathbf{n}, \mathbf{x}_{1}, \mathbf{x}_{2} \right\}

Then, xij\mathbf{x}_{ij} can be expressed as follows:

xij=aijn+bij1x1+bij2x2 \mathbf{x}_{ij} = a_{ij} \mathbf{n} + b^{1}_{ij} \mathbf{x}_{1} + b^{2}_{ij} \mathbf{x}_{2}

These coefficients bij1,bij2b_{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:

⟨xij,xl⟩= bij1⟨x1,xl⟩+bij2⟨x2,xl⟩= ∑k′=12bijk′⟨xk′,xl⟩= ∑k′=12bijk′gk′l  ⟹  ∑l=12⟨xij,xl⟩glk= ∑l=12∑k′=12bijk′gk′lglk= ∑k′=12bijk′δk′k= bijk \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 bijkb_{ij}^{k} as Γijk\Gamma_{ij}^{k} and define it as follows:

Definition

The Γijk(1≤i,j,k≤2)\Gamma_{ij}^{k}(1\le i,j,k \le 2) defined as follows is called the Christoffel symbol.

Γijk:=∑l=12⟨xij,xl⟩glk=⟨xij,xl⟩glk \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 x12=x21\mathbf{x}_{12} = \mathbf{x}_{21}, it follows that Γ12k=Γ21k\Gamma_{12}^{k} = \Gamma_{21}^{k}.

The tangent components bijkb_{ij}^{k} of xij\mathbf{x}_{ij} are denoted as Γijk\Gamma_{ij}^{k} and called Christoffel symbols, and the normal component aija_{ij} of xij\mathbf{x}_{ij} is denoted as LijL_{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.

Γij∣l:=∑k=12Γijkgkl \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 {ijk}\begin{Bmatrix} ij \\ k \end{Bmatrix}.

See also