Torsion Tensor - Geodesics and The Absorption of Torsion

Geodesics and The Absorption of Torsion

Suppose that γ(t) is a curve on M. Then γ is an affinely parametrized geodesic provided that

for all time t in the domain of γ. (Here the dot denotes differentiation with respect to t, which associates with γ the tangent vector pointing along it.) Each geodesic is uniquely determined by its initial tangent vector at time t=0, .

One application of the torsion of a connection involves the geodesic spray of the connection: roughly the family of all affinely parametrized geodesics. Torsion is the ambiguity of classifying connections in terms of their geodesic sprays:

• Two connections ∇ and ∇′ which have the same affinely parametrized geodesics (i.e., the same geodesic spray) differ only by torsion.

More precisely, if X and Y are a pair of tangent vectors at p ∈ M, then let

be the difference of the two connections, calculated in terms of arbitrary extensions of X and Y away from p. By the Leibniz product rule, one sees that Δ does not actually depend on how X and Y' are extended (so it defines a tensor on M). Let S and A be the symmmetric and alternating parts of Δ:

Then

• is the difference of the torsion tensors.
• ∇ and ∇′ define the same families of affinely parametrized geodesics if and only if S(X,Y) = 0.

In other words, the symmetric part of the difference of two connections determines whether they have the same parametrized geodesics, whereas the skew part of the difference is determined by the relative torsions of the two connections. Another consequence is:

• Given any affine connection ∇, there is a unique torsion-free connection ∇′ with the same family of affinely parametrized geodesics.

This is a generalization of the fundamental theorem of Riemannian geometry to general affine (possibly non-metric) connections. Picking out the unique torsion-free connection subordinate to a family of parametrized geodesics is known as absorption of torsion, and it is one of the stages of Cartan's equivalence method.