Obtaining the connection from Parallel Transport

In summary, the conversation is about obtaining the Levi-Civita connection through the concept of parallel transport. The speaker asks for help in proving this and mentions running into roadblocks, but ultimately states they have solved the problem.
  • #1
InbredDummy
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How do I obtain the Levi-Civita connection from the concept of parallel transport?

So Do Carmo asks to prove that for vector fields X, Y on M, and for c(t) an integral curve of X, i.e. c(t_0) = p and X(c(t)) = dc/dt, the covariant derivative of Y along X is the derivative of the parallel transport of Y(c(t)).

Do I just prove that the derivative of the parallel transport of a vector field satisfies the definition of an affine connection, metric compatibility and symmetric properties?

I tried doing this but I ran into some road blocks.

Is there an elegant way prove this?
 
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  • #2
I think I solved it. Thanks anyway.
 

Related to Obtaining the connection from Parallel Transport

1. What is parallel transport?

Parallel transport is a concept in differential geometry that describes how a vector or tangent space can be transported along a curve or surface without changing its direction. It is an important concept in understanding the curvature of a space.

2. How is the connection obtained from parallel transport?

The connection is obtained by considering the change in the vector or tangent space as it is transported along a curve or surface. This change is caused by the curvature of the space and is described by the Christoffel symbols, which are used to define the connection.

3. What is the significance of obtaining the connection from parallel transport?

Obtaining the connection from parallel transport allows us to understand the curvature of a space and how it affects the transport of vectors. This is important in various fields such as physics and engineering, where understanding the curvature of space is crucial in predicting the behavior of objects.

4. Can the connection be obtained for any space?

The connection can be obtained for any Riemannian space, which is a type of space that has a well-defined metric and is used in the study of differential geometry. However, for more complex spaces, the computation of the connection may be more difficult and may require advanced mathematical techniques.

5. How is parallel transport related to geodesic curves?

Parallel transport is closely related to geodesic curves, which are curves that have the property of being the shortest path between two points in a space. The connection obtained from parallel transport is used to define the geodesic equation, which describes the path that a particle will follow in a curved space with no external forces acting on it.

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