## § Christoffel symbols, geometrically

Suppose we have a manifold $M$. of dimension $d$ that has been embedded isometrically into $\mathbb R^n$. So we have a function $e: \mathbb R^d \rightarrow \mathbb R^n$ which is the embedding. We will identify $M$ to be the subspace $Im(e)$. Recall that $\partial_{x_i} e : \mathbb R^d \rightarrow \mathbb R^n$ is defined as:
\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 $\mathbb R^d \rightarrow \mathbb R^n$.
• The tangent space at point $p \in Image(e)$ is going to be spanned bythe basis $\{ \partial_{x_i}e \vert_p : \mathbb R^n \}$.
• The metric tensor of $M$,$g_{ij} \equiv \langle \frac{\partial e}{\partial x_i} \vert \frac{\partial e}{\partial x_j} \rangle$.That is, the metric tensor "agrees" with the dot product of theambient space $\mathbb R^n$.
• A vector field $V$ on the manifold $M$ is by definition a combination ofthe tangent vector fields. $V(p_0) \equiv v^j(p_0) \partial_{x_j} e(p_0)$
We can calculate the derivaive of this vector field as follows:
\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.
$(\partial_{x_i} \partial_{x_j} e )(p) \equiv \Gamma^k_{ij} \partial_{x_k} e + \vec{n}$
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 $\mathbb R^n$ onto the tangent space $T_p M$. This is defined by the equations:
\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}