Geodesic Curvature (Curvature of a curve)

Reality_Patrol

Can anyone point me to good reference that fully develops the geometry of geodesic curvature? Most of the ones I have manage to derive it, then show it's the normal to the curve, then never mention it again.

I want to know how it relates to the metric, first second or third.

Thanks.

Doodle Bob

I'm not sure what you mean by "first second or third." But, as I seem to be endlessly saying, I recommend Do Carmo's Geometry of Curves of Surfaces for a very rich discussion of curvature (geodesic curvature in particular) in 3-space and what it means. Much of the topic regarding curves in a general Riemannian manifold is similar in flavor.

As far as I know, the geodesic curvature isn't the normal of the tangent of the curvature, rather it is more or less the length of the derivative of the tangent vector, and it tells you whether the curve is instanteneously geodesic at a particular point or not.

Daverz

I think he means first and second fundamental forms. Is there a third fundamental form?

Doodle Bob

it occurs to me that you might find what you're looking for in the 2nd volume of Spivak's A Comprehensive Introduction of Differential Geometry.

In order to better answer your question, it would help to know what your objective is in studying geodesic curvature (ie what do you want to do with it?) and what resources you've looked up so far.

Reality_Patrol

That's exactly the kind of thing I'm looking for, but I'd like to see it developed in a more explicit form of course. Thanks for the references, I'm studying GR. But I've found that the geometry is clearer to me if developed in 3-space first then generalized to n-space. Thanks guys.

robphy

Possibly useful:

On the differential geometry of curves in Minkowski space
http://arxiv.org/abs/gr-qc/0601002

Semi-Riemannian Geometry With Applications to Relativity
by Barrett O'Neill
http://www.amazon.com/Semi-Riemannia.../dp/0125267401

Reality_Patrol

What publication does this paper come from? It's excellent, and I want to find more like it.

Doodle Bob

It is indeed very cool. Thanks for pointing it out.

robphy

http://arxiv.org hosts e-prints, which may or may not end up in other publications [like journals or books]. To find others like it, I'd start by searching for other articles by the authors on arxiv.org and on the web using http://scholar.google.com/ . Then, I'd search for similar topics and titles.