- #1
Jianbing_Shao
- 90
- 2
In general relativity, a vector parallel along a curve on a manifold M with a connection field Γ can be expressed:
∂v+Γv=0
We know that if the curvature corresponding to Γ is non-zero, which means if we parallel transport a vector along different paths between two points, we will get different result, so we can say we can not get a globally parallel vector field on the manifold.
Does this conclusion means that to the parallel transport equation, if curvature of Γ is not zero, then we can not find a global vector field on manifold M to satisfy the parallel equation?
∂v+Γv=0
We know that if the curvature corresponding to Γ is non-zero, which means if we parallel transport a vector along different paths between two points, we will get different result, so we can say we can not get a globally parallel vector field on the manifold.
Does this conclusion means that to the parallel transport equation, if curvature of Γ is not zero, then we can not find a global vector field on manifold M to satisfy the parallel equation?