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.

 

[Entropia]

An intrinsic derivative is a mathematical concept used in differential geometry to describe the rate of change of a geometric object without reference to an external coordinate system. It is a measure of how the object changes with respect to itself, rather than with respect to an external observer or coordinate system.

The intrinsic derivative is often referred to as the absolute derivative, as it is independent of any coordinate system. It takes into account the intrinsic properties of the object, such as its curvature and shape, and is not influenced by the way it is embedded in a larger space.

On the other hand, the covariant derivative is a derivative that takes into account both the intrinsic properties of the object and the external coordinate system it is embedded in. It is a more general concept that can be applied to objects that are not necessarily defined on a flat space, such as curved surfaces.

The geometric interpretation of the intrinsic derivative is that it describes the rate of change of a geometric object as it moves along its own surface. It can be thought of as a way to measure the object's "internal" changes, rather than its changes in relation to an external observer.

In summary, the main difference between the intrinsic and covariant derivatives is that the former is independent of any external coordinate system, while the latter takes into account both the intrinsic and extrinsic properties of the object. The intrinsic derivative has a more "internal" interpretation, while the covariant derivative combines both internal and external changes.