Riemann Tensor, Stoke's Theorem & Winding Number

[jk22]

I saw briefly that the Riemann tensor can be obtained via Stoke's theorem and parallel transport along a closed curve.

If one does add winding number then it can give several results, does it imply that this tensor is multivalued ?

 

[PeterDonis]

I saw briefly

Where? Please give a reference.

 

[martinbn]

I saw briefly that the Riemann tensor can be obtained via Stoke's theorem and parallel transport along a closed curve.

If one does add winding number then it can give several results, does it imply that this tensor is multivalued ?

No, because winding around a closed curve several times is a different closed curve.

 

[jk22]

Then how is the coming back at the same time as starting, it should but start again or time were stopped ?

Namely if a vector is parallel transported along a CTC is it coming back to its original state ?

I think I'm locked since the two alternatives are for me : If it is yes then there is no curvature, if it is no then the vector at $(x^\mu)$ has several values, hence multivalued ?

What am I thinking wrong here ?

 

[PeterDonis]

if a vector is parallel transported along a CTC

Where did CTCs come into it?

 

[PAllen]

Then how is the coming back at the same time as starting, it should but start again or time were stopped ?

Namely if a vector is parallel transported along a CTC is it coming back to its original state ?

I think I'm locked since the two alternatives are for me : If it is yes then there is no curvature, if it is no then the vector at $(x^\mu)$ has several values, hence multivalued ?

What am I thinking wrong here ?

I am going to guess what might be confusing you (always dangerous, but I'll risk it). I'm thinking you have in mind a vector field with a value at every spacetime point (event). You take one of these and parallel transport around a circuit. You get a different vector. Around a circuit again, you get still a different vector. But each of these transported vectors is a different vector than the original, not part of the original vector field at all. There is no multi-valued anything. Curvature is defined via a limiting operation for circuits that return to the starting event exactly once. Note, CTC's are not special at all, in this. In defining curvature at a point, you will consider spacelike circuits and timelike circuits, even mixed. It doesn't matter for defining curvature.