Covariant derivative from connections

Thread starter lavinia
Start date Oct 22, 2011
Tags

Covariant Covariant derivative Derivative

In summary, a covariant derivative is a mathematical concept used in differential geometry and tensor calculus to describe how a vector field changes along a curve or surface. It takes into account the curvature and connection of the underlying space. A connection is a mathematical structure used to define how vectors at different points on a manifold can be compared and related to each other. The covariant derivative is important because it allows us to study vector fields on curved spaces, which is essential in physics and mathematics. It is calculated using the connection, which is defined by a set of coefficients known as Christoffel symbols derived from the metric tensor. The covariant derivative differs from the partial derivative, as it takes into account the curvature of the space, while the partial derivative is

Oct 22, 2011

lavinia

Science Advisor

Gold Member

On a 2 dimensional Riemannian manifold how does one derive the covariant derivative from the connection 1 form on the tangent unit circle bundle?

Physics news on Phys.org

Oct 22, 2011

Ben Niehoff

Science Advisor

Gold Member

Its the same formula in any dimension. Let [itex]X = X^a e_a[/itex] be a vector field, where [itex]e_a[/itex] is an orthonormal frame. Then

[tex]\nabla X = (D X^a) \otimes e_a = (d X^a + \omega^a{}_b X^b) \otimes e_a[/tex]

Related to Covariant derivative from connections

What is a covariant derivative?

A covariant derivative is a mathematical concept used in differential geometry and tensor calculus to describe how a vector field changes along a curve or surface. It takes into account the curvature and connection of the underlying space.

What is a connection in the context of covariant derivatives?

A connection is a mathematical structure used to define how vectors at different points on a manifold (a space with variable curvature) can be compared and related to each other. It is a way of extending the notion of differentiation from flat spaces to curved spaces.

Why is the covariant derivative important?

The covariant derivative allows us to define and study the behavior of vector fields on curved spaces, which is essential in many areas of physics and mathematics. It also plays a crucial role in the formulation of various physical theories, such as general relativity.

How is the covariant derivative calculated?

The covariant derivative is calculated using the connection, which is defined by a set of coefficients known as Christoffel symbols. These symbols are derived from the metric tensor, which describes the curvature of the underlying space.

What is the relationship between the covariant derivative and the partial derivative?

The partial derivative is defined on flat spaces and does not take into account the curvature of the space. In contrast, the covariant derivative is defined on curved spaces and takes into account the connection and curvature. The partial derivative can be seen as a special case of the covariant derivative on flat spaces.