How Does Intrinsic Derivative Differ from Covariant Derivative in Geometry?

[off-diagonal]

What is it? and How different between intrinsic(absolute) derivative and covariant derivative?

What is its geometric interpretation?

 

[atyy]

If on a manifold you have two vector fields V and W, then you can take the covariant derivative of V wrt W at every point in the manifold. The vector field W can also be thought of as a family of curves whose tangent vectors form W. The absolute derivative of V at any point on a particular curve in that family is the covariant derivative of V wrt W at that point in the curve.

The geometric idea is that in a vector space, there is an idea of two vectors being parallel. But on a manifold, there is a vector space at each point in the manifold, but there is no predefined notion of vectors at different points on a manifold being parallel. The covariant derivative/absolute derivative defines the notion of "parallel transport" that allows you to say if vectors at two nearby points on a curve are parallel or not.

I checked the definition on p377 of http://books.google.com/books?id=vQ...D+physics+geometry+liek&source=gbs_navlinks_s.