It's much easier to "visualize" the covariant
derivative using a higher dimensional Euclidean "scaffolding" into which
you isometrically imbed your manifold (always possible, at least for spaces
with a positive definite metric, but in anycase, the results are the same).
By the way, it's MUCH easier to derive most of the standard diff geo formulas
If you have a continuous, regular, C_infinity, blah,
blah, blah manifold (think smooth curved surface) with a vector field defined
over it. Take the (everday, ol' Euclidean vector) derivative of this field
with respect to some parameter. The result will be a new vector field
that does not necessarily reside in the tangent space of the manifold.
The PROJECTION of the derivative vector onto the tangent
space of the manifold is the covariant derivative. It is the
derivative field with the normal component subtracted off. It is
the component of the everyday ol' Euclidean derivative that resides in
the tangent space of the manifold. Think "shadow" of the derivative vector
with the light directly overhead. INTRINSICALLY, the normal component
doesn't really exist. EXTRINSICALLY, it is not unique as it depends
on the imbedding, which is not unique. The covariant derivative IS
unique and does not depend on the (isometric) imbedding.
Now if you do this twice (take the second covariant derivative),
then do it again, but reverse the order, then subtract, you get the Riemann
Curvature Tensor. ie:
- vi;k;j = -v m R imjk
Where vi is the indexed vector
field and ";j " and ";k " are the covariant derivatives of
v with respect to parameters indexed by j and k,
and repeated indices are summed. Depending on your bent, the above
can serve as a definition for the curvature tensor. In flat space
R is of course zero as the second derivative commutes.