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Geometric relevance: The torsion τ(s) measures the turnaround of the binormal vector. The larger the torsion is, the faster the binormal vector rotates around the axis given by the tangent vector (see graphical illustrations). In the animated figure the rotation of the binormal vector is clearly visible at the peaks of the torsion function.
The torsion tensor is a bilinear map of two input vectors ,, that produces an output vector (,) representing the displacement within a tangent space when the tangent space is developed (or "rolled") along an infinitesimal parallelogram whose sides are ,.
On the example of a torus knot, the tangent vector T, the normal vector N, and the binormal vector B, along with the curvature κ(s), and the torsion τ(s) are displayed. At the peaks of the torsion function the rotation of the Frenet–Serret frame (T,N,B) around the tangent vector is clearly visible.
The torsion submodule of a module is the submodule formed by the ... any free abelian group is torsion-free and any vector space over a field K is torsion-free when ...
the tangent vector at point p of the i th coordinate curve. The ∂ i are a natural basis for the tangent space at point p, and the X i the corresponding coordinates for the vector field X = X i ∂ i. When both curvature and torsion vanish, the connection defines a pre-Lie algebra structure on the space of global sections of the tangent bundle.
To describe the torsion, note that the vector bundle E is the tangent bundle. This carries a canonical solder form (sometimes called the canonical one-form , especially in the context of classical mechanics ) that is the section θ of Hom(T M , T M ) = T ∗ M ⊗ T M corresponding to the identity endomorphism of the tangent spaces.
for all vector fields X, Y and Z on M, where ((,)) denotes the derivative of the function (,) along this vector field . The Levi-Civita connection is the torsion-free Riemannian connection on a manifold.
Analytic torsion (or Ray–Singer torsion) is an invariant of Riemannian manifolds defined by Daniel B. Ray and Isadore M. Singer (1971, 1973a, 1973b) as an analytic analogue of Reidemeister torsion. Jeff Cheeger ( 1977 , 1979 ) and Werner Müller ( 1978 ) proved Ray and Singer's conjecture that Reidemeister torsion and analytic torsion are the ...