Relation between Frenet-Serret torsion and the torsion tensor?

In summary, The conversation discusses the relationship between Frenet-Serret curvature and Riemann curvature of a manifold and the potential relationship between Frenet-Serret torsion and the torsion tensor for a general manifold. The source provided is recommended for further reading on the topic.
  • #1
lucyk
2
0
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
 
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  • #2
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
 
  • #3
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
 

What is the Frenet-Serret torsion?

The Frenet-Serret torsion is a mathematical concept that measures the rate of change of the direction of a curve in three-dimensional space. It is a key component in the study of curves and their properties, particularly in the field of differential geometry.

What is the torsion tensor?

The torsion tensor is a mathematical object that describes the twisting or rotational deformation of a three-dimensional space. It is closely related to the Frenet-Serret torsion, but it is defined for general curves and not just for space curves.

How are the Frenet-Serret torsion and the torsion tensor related?

The Frenet-Serret torsion is a special case of the torsion tensor, specifically for space curves. The torsion tensor can be calculated using the Frenet-Serret formulas, which involve the curvature and torsion of the curve. Essentially, the Frenet-Serret torsion is a component of the torsion tensor.

What is the significance of the relation between Frenet-Serret torsion and the torsion tensor?

The relation between these two concepts allows for a deeper understanding of the properties of curves and their behavior in three-dimensional space. It also provides a useful tool for calculating and analyzing the torsion of more complex curves using the Frenet-Serret formulas.

Are there any real-world applications of the relation between Frenet-Serret torsion and the torsion tensor?

Yes, there are many real-world applications of this relation in fields such as computer graphics, robotics, and physics. For example, it is used in computer-aided design to model and manipulate 3D objects, and in robotics to calculate the torsional forces on joints and structures. It is also relevant in the study of fluid dynamics and elasticity.

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