Relation between Frenet-Serret torsion and the torsion tensor?

[lucyk]

Hi, I was wondering if anyone could help me with this differential geometry question I've been struggling to find information on.

I (at least very roughly) understand the relationship between the Frenet-Serret curvature of a curve and the Riemann curvature of a general n-dimensional manifold: the curvature tensor is determined by the sectional curvatures of 2-D slices through the manifold, and Gauss's theorem relates these sectional curvatures to the curvature of curves along the two principal directions of the 2-D surface.

What I was wondering was is there a similar geometric relationship between the Frenet-Serret torsion of curves and the torsion tensor for a general manifold, and if so are there any good sources for reading about it?

Thanks,
Lucy

 

[lucyk]

ah beginning to see where to start looking. I have Aminov's 'Geometry of Submanifolds' and may also brave Spivak volume 3 or 4 for more on submanifolds... any other suggestions welcome

 

[gato_]

Despite the name, both curvature and torsion of the curve are related to the curvature. I particularly like the approach with differential forms
http://arxiv.org/PS_cache/arxiv/pdf/0706/0706.0001v2.pdf
but it is probably the best one to start with