## Geodesic Curvature (Curvature of a curve)

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.

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 Quote by 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.
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.

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

## Geodesic Curvature (Curvature of a curve)

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.

 Quote by Doodle Bob 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.
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.

 Blog Entries: 47 Recognitions: Gold Member Homework Help Science Advisor 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

 Quote by robphy Possibly useful: On the differential geometry of curves in Minkowski space http://arxiv.org/abs/gr-qc/0601002
What publication does this paper come from? It's excellent, and I want to find more like it.

 Quote by Reality_Patrol What publication does this paper come from? It's excellent, and I want to find more like it.
It is indeed very cool. Thanks for pointing it out.

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