How Does Intrinsic Derivative Differ from Covariant Derivative in Geometry?

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The intrinsic (absolute) derivative and covariant derivative are fundamental concepts in differential geometry, particularly when analyzing vector fields on manifolds. The covariant derivative allows for the differentiation of vector fields along curves, providing a framework for defining parallel transport. This is crucial because, unlike in vector spaces, there is no inherent notion of parallelism between vectors at different points on a manifold. The geometric interpretation emphasizes the relationship between vector fields and their behavior along curves within the manifold.

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

What is its geometric interpretation?
 
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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.
 
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