Parallel Transport

The notion of parallel transport on a manifold M makes precise the idea of translating a vector field V along a differentiable curve to attain a new vector field V^' which is parallel to V. More precisely, let M be a smooth manifold with affine connectionVector Bundle Connection del , let c:I->M be a differentiable curve from an interval I into M, and let V_0 in T_(c(t_0))M be a vector tangent to M at c(t_0) for some t_0 in I. A vector field V is said to be the parallel transport of V_0 along c provided that V(t), t in I, is a vector field for which V(t_0)=V_0.

Note that the use of the quantifier parallel in the above definition makes reference to the fact that a parallel transport V(t) of a vector field V_0 along a curve c is necessarily covariantly constant, i.e., V satisfies


for all t in I where, here, DV/dt denotes the unique covariant derivative of V associated to del .

A standard result in differential geometry is that, under the above hypotheses, parallel transports are unique.

In addition to the above definition, some literature defines parallel transport in a more function analytic way. Indeed, given an interval I and a point t in I, a parallel transport V(t) of V_0 along c:I->M is nothing more than a linear transformation


which maps V_0 to V(t). It is obvious that this transformation is invertible, its inverse being given simply by parallel transport along the reversed portion of c from t to t_0. The expression tau_t has added benefit, too, because despite being defined intrinsically in terms of the affine connection del on M, it also provides a mechanism whereby one can recover a manifold's affine connection given a collection V_1,...,V_n of parallel vector fields along a curve c. In particular, if c(0)=p and c^'(0)=X_p, then

 del _(X_p)Y=lim_(h->0)(tau_h^(-1)Y_(c(h))-Y_p)/h

where del _(X_p)Y=del _XY(p) is the desired vector field given by the connection del and where Y_(c(h)):=Y(c(h)).

See also

Covariant Derivative, Manifold, Parallel Vectors, Vector field

This entry contributed by Christopher Stover

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Do Carmo, M. Riemannian Geometry. Boston, MA: Birkhäuser, 1993.Spivak, M. A Comprehensive Introduction to Differential Geometry, Vol. 2, 3rd ed. Berkeley, CA: Publish or Perish Press, 1999.

Cite this as:

Stover, Christopher. "Parallel Transport." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein.

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