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:

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 by the 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 the ambient space $\mathbb R^n$.
• A vector field $V$ on the manifold $M$ is by definition a combination of the 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:

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.

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:

§ References

• The wikipedia page on Covariant derivative