Relation between affine connection and covariant derivative

[nilsgeiger]

I now study general relativity and have a few questions regarding the mathematical formulation:

1) What ist the relation between an connection and a covariant derivative?
Can you explain the exact difference?

2) One a lorentzian manifold, what ist the relation between the levi-civita-connection and the used covariant derivative?

Unrelated, but probably not important enough to justify a new thread:

3) In the einstein field equitations the ricci curvature tensor is used, the non trivial contraction of the riemann curvature tensor.
My professor told me, you can understand this as averaging along all curves of the parellel transports.
Is there an easy way to see this?

 

[andrewkirk]

1) What ist the relation between an connection and a covariant derivative?
Can you explain the exact difference?

The affine connection is the operator $\nabla$ that takes a vector and a vector field as input and gives a vector as result. A covariant derivative is the result $\nabla_XY$ of applying that operator to a vector field $Y$ and a vector $X$.

At least, that's the definition used in my text: John Lee's 'Riemannian Manifolds: an Introduction to Curvature'. I daresay the exact definitions vary between writers.

 

[andrewkirk]

2) One a lorentzian manifold, what ist the relation between the levi-civita-connection and the used covariant derivative?

If one follows the framework of my previous post, the Levi-Civita connection on a Pseudo-Riemannian manifold is the affine connection that is both symmetric and compatible with the metric. 'Compatible' has a number of different definitions, the simplest of which is that the metric must have zero covariant derivative under that connection, in every direction, everywhere in the manifold (often written as $\nabla g\equiv 0$).

It is provable that the Levi-Civita connection is unique, ie there is only one affine connection on a Pseudo Riemannian manifold that has those two properties.

The covariant derivative of a vector field on a Pseudo-R manifold with respect to a vector in its tangent bundle will be the result of applying the Levi-Civita connection to that vector field and vector.