Covariant derivative repect to connection?

[noomz]

Could anyone tell me about the covariant derivative with respect to the connection?

 

[Fredrik]

Let V be the set of smooth vector fields on a manifold M. A connection is a map \nabla:V\times V\rightarrow V such that

(i) \nabla_{fX+gY}Z=f\nabla_X Z+g\nabla_YZ

(ii) \nabla_X(Y+Z)=\nabla_XY+\nabla_XZ

(iii) \nabla_X(fY)=(Xf)Y+f\nabla_XY

The map \nabla_XY is the covariant derivative of Y in the direction of X. The covariant derivative operator corresponding to a coordinate system x is

\nabla_{\frac{\partial}{\partial x^\mu}

The notation is often simplified to

\nabla_{\partial_\mu}

or just \nabla^\mu.

That might be enough to get you started, but maybe I just told you what you already know. You might want to check out the Wikipedia articles with titles "affine connection", "levi-civita connection", "parallel transport" and "covariant derivative".

 

[noomz]

I'm working on a general relativity project, about Palatini formalism. I've been studying differential geometry for a while in order to understand covariant derivative and connection. What you've just told me I've read it a little but I don't understand enough. To understand covariant derivative with repect to connection what do I need to know? In general, covariant derivative respect to what? Now I'm so confused. hahaha ^^

 

[mma]

I think that the connection you mean is the Levi-Civita connection of General Relativity.
The Levi-Civita connection is a special connection defined on the tangent bundle of a Riemannian manifold. To understand the general concept of connection you have to know fiber bundles. Fiber bundles are similar to Cartesian product with the difference that they have only one canonical projection instead of two. Think for example of a beehive hairstyle like of Marge Simpson from The Simpsons. This is a fiber bundle with a 3-dimensional total space and a 2-dimensional base space. The total space is the hairdo itself while the base space is the skin of the head. The fibers are the hairs. The projection projects each point inside the hairdo to the skin along the hair present at this point. But other projection doesn't exist. Each hair is regarded as a "vertical" line, but "horizontal" lines don't exist except you have a connection. Connection is an assignment of the "horizontal" directions to each point. In this special case, you have at least two different "natural" choices. One is that you regard a curve in the hairdo horizontal if it is in a horizontal plane of our 3-dimensional space, while the other possibility that you regard a curve in the hairdo horizontal, if it meets every hair at the same length. A connection defines the so-called "horizontal lift" as follows. If you draw a curve on the base space (i.e, on the skin of the head), and choose a point in the bundle above this curve (i.e, in the hairdo on a hair growing out from a point of the given curve), then you can define the horizontal lift of this curve to the point chosen. This is a curve that passes through the point, its direction is horizontal everywhere and its projection to the base space is the given curve.

In the case of a tangent bundle, the manifold M plays the role of the skin of the head (base space) and the tangent spaces T_pM plays the role of the hairs (fibers). The projection projects every tangent vector in T_pM to the point p of M. This is the vertical projection. But horizontal projection doesn't exist except you have a connection. A connection defines the horizontal lift of a curve say c(t) on M to the tangent bundle say v(t). Of course, v(t) is in T_c(t)M for every t. In the case of the tangent bundle we say that v(t) is "parallel" to v(0). So, instead of saying that "v(t) is the point above the point c(t) on the horizontal lift of c to v(0)" we say that "v(t) is the parallel transport of the vector v(0) along c from c(0) to c(t)". Now, the covariant derivative of the vector field Y in the direction X is

<br /> \nabla_X(Y)=\lim_{h \rightarrow 0}\frac{1}{h}\left[\tau^h(Y(c(h)) - Y(c(0))\right]<br />

where c is a curve in M having \dot c(0) = X and \tau^h(Y(c(h)) is the parallel transport of Y(c(h)) to c(0) along c. Of course this derivative depends on the parallel transporting function, i.e. on the connection.