§ Christoffel symbols, geometrically
Suppose we have a manifold . of dimension that has been embedded isometrically
into . So we have a function
which is the embedding. We will identify to be the subspace .
is defined as:
Note that it is a function of type .
We can calculate the derivaive of this vector field as follows:
- The tangent space at point is going to be spanned by the basis .
- The metric tensor of , . That is, the metric tensor "agrees" with the dot product of the ambient space .
- A vector field on the manifold is by definition a combination of the tangent vector fields.
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
This gives us the Christoffel symbols as "variation of second derivative along
§ 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 onto
the tangent space . This is defined by the equations: