The wikipedia isn't too bad: let me summarize briefly what I learned from 5 minutes reading having previously known dirt all about (Psuedo)-Riemannian geometry:
(Original post by shawrie777)
I'm trying to understand different types of geodesics. I understand that a geodesic is the shortest distance between two points on a manifold, but what's are these types:
metric - I assume this is just the distance as calculated from the metric tensor
Can someone explain these?
A (metric) geodesic on a Riemannian manifold is loosely a curve that is locally the shortest distance between two points.
If you have a smooth manifold with an affine connection then one can define a geodesic of a curve in terms of parallel transport. On a Riemannian manifold, if you take the Levi-Civita connection and solve the geodesic equations then you can show that a geodesic in this sense is the same as a geodesic in the sense of locally minimising the metric.
The last three adjectives, as one would expect, refer to geodesics on a Lorentzian manifold. One may also equip a Lorentzian manifold with the Levi-Civita connection which is the unique connection satisfying the same properties as in the Riemannian case. Thus again there is no distinction between metric geodesics and affine geodesics w.r.t Levi-Civita. Then (assuming the positive part of the metric is space and the negative part is time) a geodesic is space like if the norm of its tangent vector is positive, timelike if the norm of its tangent vector is negative and null if it is zero.
Last edited by Jake22; 17-04-2012 at 17:12.