I recently derived the Riemann tensor (R(adsbygoogle = window.adsbygoogle || []).push({}); ^{a}_{bmv}) for the 2 sphere.

I then did R^{a}_{bmv}U^{b}V^{m}W^{v}to calculate dv^{a}(the change in the vector v^{a}as you parallel transport it around the loop of the sphere).

The result I got for dv^{1}was 0. I got 0 for dv^{2}as well.

I am just making this thread to verify if I am correct in getting 0. Should I get a change of <0,0> if I parallel transport a vector around a loop on the 2D surface of a sphere?

Also, for the curvature scalar R, I got 2/r^{2}. What info exactly does the curvature scalar give you besides telling you whether or not a space is curved?

Here was the metric tensor I used for all of this:

g_{11}= r^{2}

g_{12}and g_{21}= 0

g_{22}= r^{2}sin^{2}(θ)

where r is actually a constant, x^{1}is θ and x^{2}is ø

**Physics Forums | Science Articles, Homework Help, Discussion**

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Parrallel transport on the 2-sphere

Loading...

Similar Threads for Parrallel transport sphere |
---|

I Calculating the Ricci tensor on the surface of a 3D sphere |

I What is the "limit of vanishing transport velocity"? |

A Global solution to parallel transport equation? |

I Lie Derivatives vs Parallel Transport |

I Parallel transport vs Fermi Transport |

**Physics Forums | Science Articles, Homework Help, Discussion**