§ Christoffel symbols, geometrically

Suppose we have a manifold MM. of dimension dd that has been embedded isometrically into Rn\mathbb R^n. So we have a function e:RdRne: \mathbb R^d \rightarrow \mathbb R^n which is the embedding. We will identify MM to be the subspace Im(e)Im(e). Recall that xie:RdRn\partial_{x_i} e : \mathbb R^d \rightarrow \mathbb R^n is defined as:
xie:RdRn[xie](p)limδx0e(p+(x0=0,x1=0,xi=δx,,xn=0))e(p)δx \begin{aligned} &\partial_{x_i}e : \mathbb R^d \rightarrow \mathbb R^n \\ &[\partial {x_i}e](p) \equiv \lim_{\delta x \rightarrow 0} \frac{e(p + (x_0=0, x_1=0\dots, x_i=\delta_x, \dots, x_n=0)) - e(p)}{\delta x} \end{aligned}
Note that it is a function of type RdRn\mathbb R^d \rightarrow \mathbb R^n.
  • The tangent space at point pImage(e)p \in Image(e) is going to be spanned by the basis {xiep:Rn}\{ \partial_{x_i}e \vert_p : \mathbb R^n \}.
  • The metric tensor of MM, gijexiexjg_{ij} \equiv \langle \frac{\partial e}{\partial x_i} \vert \frac{\partial e}{\partial x_j} \rangle. That is, the metric tensor "agrees" with the dot product of the ambient space Rn\mathbb R^n.
  • A vector field VV on the manifold MM is by definition a combination of the tangent vector fields. V(p0)vj(p0)xje(p0)V(p_0) \equiv v^j(p_0) \partial_{x_j} e(p_0)
We can calculate the derivaive of this vector field as follows:
V(p)xi=xi[vj(p)xje]=vjxixje+xjexivj \begin{aligned} &\frac{V(p)}{\partial x^i} = \partial_{x_i} \left[ v^j(p) \partial_{x_j} e \right] \\ &= v^j \cdot \partial_{x_i} \partial_{x_j} e + \partial_{x_j}e \cdot \partial_{x_i} v^j \end{aligned}
We choose to rewrite the second degree term in terms of the tangent space, and some component that is normal to us that we have no control over.
(xixje)(p)Γijkxke+n (\partial_{x_i} \partial_{x_j} e )(p) \equiv \Gamma^k_{ij} \partial_{x_k} e + \vec{n}
This gives us the Christoffel symbols as "variation of second derivative along the manifold.

§ Relationship to the Levi-Cevita connection

The covariant derivative defined by the Levi-Cevita connection is the derivative that contains the projection of the full derivative in Rn\mathbb R^n onto the tangent space TpMT_p M. This is defined by the equations:
eiVxiVn=Πn[vjxixje+xjexivj]=Πn[vj(Γijkxke+n)+xjexivj]=vj(Γijkxke+0)+xjexivj=vj(Γijkxke+0)+xkexivk=vjΓijkxke+xkexivk=xke(vjΓijk+xivk) \begin{aligned} &\nabla_{e_i} V \equiv \partial_{x_i} V - \vec{n} \\ &= \Pi_{\vec{n}^\bot} \left [v^j \cdot \partial_{x_i} \partial_{x_j} e + \partial_{x_j}e \cdot \partial_{x_i} v^j \right] \\ &= \Pi_{\vec{n}^\bot} \left[ v^j \cdot (\Gamma^k_{ij} \partial_{x_k} e + \vec{n})+ \partial_{x_j}e \cdot \partial_{x_i} v^j \right] \\ &= v^j \cdot (\Gamma^k_{ij} \partial_{x_k} e + \vec 0) + \partial_{x_j}e \cdot \partial_{x_i} v^j \\ &= v^j \cdot (\Gamma^k_{ij} \partial_{x_k} e + \vec 0) + \partial_{x_k}e \cdot \partial_{x_i} v^k \\ &= v^j \cdot \Gamma^k_{ij} \partial_{x_k} e + \partial_{x_k}e \cdot \partial_{x_i} v^k \\ &= \partial_{x_k} e \left( v^j \cdot \Gamma^k_{ij} + \partial_{x_i} v^k \right) \\ \end{aligned}

§ References