Geometric Calculus - Covariant Derivative

Covariant Derivative

A sufficiently smooth k-surface in an n-dimensional space is deemed a manifold. To each point on the manifold, we may attach a k-blade B that is tangent to the manifold. Locally, B acts as a pseudoscalar of the k-dimensional space. This blade defines a projection of vectors onto the manifold:

Just as the geometric derivative is defined over the entire n-dimensional space, we may wish to define an intrinsic derivative, locally defined on the manifold:

If a is a vector tangent to the manifold, then indeed both the geometric derivative and intrinsic derivative give the same directional derivative:

Although this operation is perfectly valid, it is not always useful because itself is not necessarily on the manifold. Therefore we define the covariant derivative to be the forced projection of the intrinsic derivative back onto the manifold:

Since any general multivector can be expressed as a sum of a projection and a rejection, in this case

we introduce a new function, the shape tensor, which satisfies

where is the commutator product. In a local coordinate basis spanning the tangent surface, the shape tensor is given by

Importantly, on a general manifold, the covariant derivative does not commute. In particular, the commutator is related to the shape tensor by

Clearly the term is of interest. However it, like the intrinsic derivative, is not necessarily on the manifold. Therefore we can define the Riemann tensor to be the projection back onto the manifold:

Lastly, if F is of grade r, then we can define interior and exterior covariant derivatives as

and likewise for the intrinsic derivative.