User:Silly rabbit/Sandbox/Parallel transport



In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent bundle), then this connection allows one to transport vectors of the manifold along curves so that they stay parallel with respect to the connection. Other notions of connection come equipped with their own parallel transportation systems as well. For instance, a Koszul connection in a vector bundle also allows for the parallel transport of vectors in much the same way as with a covariant derivative. An Ehresmann or Cartan connection supplies a lifting of curves from the manifold to the total space of a principal bundle. Such curve lifting may sometimes be thought of as the parallel transport of reference frames.

The parallel transport for a connection thus supplies a way of, in some sense, moving the local geometry of a manifold along a curve: that is, of connecting the geometries of nearby points. There may be many notions of parallel transport available, but a specification of one &mdash; one way of connecting up the geometries of points on a curve &mdash; is tantamount to providing a connection. In fact, the usual notion of connection is the infinitesimal analog of parallel transport. Or, vice versa, parallel transport is the local realization of a connection.

As parallel transport supplies a local realization of the connection, it also supplies a local realization of the curvature known as holonomy. The Ambrose-Singer theorem makes explicit this relationship between curvature and holonomy.

Parallel transport on a vector bundle
Let M be a smooth manifold. Let E&rarr;M be a vector bundle with a connection &nabla; and &gamma;: I&rarr;M a smooth curve parameterized by an open interval I. A section $$X$$ of $$E$$ along &gamma; is called parallel if
 * $$\nabla_{\dot\gamma(t)}X=0$$, for all t in I.

Suppose we are given an element e0 &isin; EP at P = &gamma;(0) &isin; M, rather than a section. The parallel transport of e0 along &gamma; is the extension of e0 to a parallel section X on &gamma; More precisely, X is the unique section of E along &gamma; such that Note that in a local trivialization (1) defines an ordinary differential equation, with the initial condition given by (2). Thus the Picard–Lindelöf theorem guarantees the existence and uniqueness of the solution.
 * 1) $$\nabla_{\dot{\gamma}} X = 0$$
 * 2) $$X_{\gamma(0)} = e_0.$$

Thus the connection &nabla; defines a way of moving elements of the fibers along a curve, and this provides linear isomorphisms between the fibers at points along the curve:
 * $$\Gamma(\gamma)_s^t : E_{\gamma(s)} \rightarrow E_{\gamma(t)}$$

from the vector space lying over &gamma;(s) to that over &gamma;(t). This isomorphism is known as the parallel transport map associated to the curve.

The isomorphisms between fibers obtained in this way will in general depend on the choice of the curve: if they do not then parallel transport along every curve can be used to define parallel sections of E over all of (the universal cover of) M. This is only be possible the curvature of &nabla; is zero.

In particular, parallel transport around a closed curve starting at a point x defines an automorphism of the tangent space at x which is not necessarily trivial. The parallel transport automorphisms defined by all closed curves based at x form a transformation group called the holonomy group of &nabla; at x. There is a close relation between this group and the value of the curvature of &nabla; at x; this is the content of the Ambrose-Singer holonomy theorem.

Recovering the connection from the parallel transport
Classically, a connection was given in terms of the parallel transport maps rather than as a differential operator. A connection, in this sense, is an assignment to each curve &gamma; in the manifold a collection of mappings
 * $$\Gamma(\gamma)_t^s : E_{\gamma(s)} \rightarrow E_{\gamma(t)}$$

such that The notion of smoothness in condition 3. is somewhat difficult to pin down (see the discussion below of parallel transport in fibre bundles). In particular, modern authors such as Kobayashi and Nomizu generally view the parallel transport of the connection as coming from a connection in some other sense, where smoothness is more easily expressed.
 * 1) $$\Gamma(\gamma)_s^s = Id$$, the identity transformation of E&gamma;(s).
 * 2) $$\Gamma(\gamma)_t^u\circ\Gamma(\gamma)_u^s = \Gamma(\gamma)_t^s$$.
 * 3) The dependence of &Gamma; on &gamma;, s, and t is "smooth."

Nevertheless, given such a rule for parallel transport, it is possible to recover the associated infinitesimal connection in E as follows. Let &gamma; be a differentiable curve in M with initial point &gamma;(0) and initial tangent vector X = &gamma;&prime;(0). If V is a section of E over &gamma;, then let
 * $$\nabla_X V = \lim_{h\to 0}\frac{\Gamma(\gamma)_0^hV_{\gamma(h)} - V_{\gamma(0)}}{h} = \left.\frac{d}{dt}\Gamma(\gamma)_0^tV_{\gamma(t)}\right|_{t=0}.$$

This defines the associated infinitesimal connection &nabla; on E. Conversely, one recovers the same parallel transport &Gamma; from this infinitesimal connection.

Special case: The tangent bundle
Let M be a smooth manifold. Then a connection on the tangent bundle of M, called an affine connection, distinguishes a class of curves called (affine) geodesics. A smooth curve &gamma;: I &rarr; M is an affine geodesic if $$\dot\gamma$$ is parallel transported along $$\gamma$$, that is
 * $$\Gamma(\gamma)_s^t\dot\gamma(s) = \dot\gamma(t).$$

Taking the derivative with respect to time, this takes the more familiar form
 * $$\nabla_{\dot\gamma(t)}\dot\gamma = 0$$.

Parallel transport in Riemannian geometry
In (pseudo) Riemannian geometry, a metric connection is any connection whose parallel transport mappings preserve the metric tensor. Thus a metric connection is any connection &Gamma; such that, for any two vectors X, Y &isin; T&gamma;(s)
 * $$\langle\Gamma(\gamma)_s^tX,\Gamma(\gamma)_s^t\rangle_{\gamma(t)}=\langle X,Y\rangle_{\gamma(s)}.$$