Parallel transport

Let M be a smooth manifold. For each point , there is an associated vector space T_pM called the tangent space of M at p. Vectors in T_pM are thought of as the vectors tangent to M at p. A Riemannian metric g on M assigns to each p a positive-definite inner product g_p : T_pM \times T_pM \to \mathbf R in a smooth way. A smooth manifold M endowed with a Riemannian metric g is a Riemannian manifold, denoted . Let x^1,\ldots,x^n denote the standard coordinates on \mathbf R^n. The Euclidean metric g^\text{euc} is given by : g^\text{euc} = (dx^1)^2 + \cdots + (dx^n)^2. Euclidean space is the Riemannian manifold (\mathbf R^n,g^\text{euc}). In Euclidean space, all tangent spaces are canonically identified with each other via translation, so it is easy to move vectors from one tangent space to another. Parallel transport of tangent vectors is a way of moving vectors from one tangent space to another along a curve in the setting of a general Riemannian manifold. Note that while the vectors are in the tangent space of the manifold, they might not be in the tangent space of the curve they are being transported along. An affine connection on a Riemannian manifold is a way of differentiating vector fields with respect to other vector fields. A Riemannian manifold has a natural choice of affine connection called the Levi-Civita connection. Given a fixed affine connection on a Riemannian manifold, there is a unique way to do parallel transport of tangent vectors. Different choices of affine connections will lead to different systems of parallel transport. Precise definition Let M be a manifold with an affine connection . Then a vector field X is said to be parallel if for any vector field , . Intuitively speaking, parallel vector fields have all their derivatives equal to zero and are therefore in some sense constant. By evaluating a parallel vector field at two points x and , an identification between a tangent vector at x and one at y is obtained. Such tangent vectors are said to be parallel transports of each other. More precisely, if \gamma:I\rightarrow M is a piecewise continuously differentiable curve parametrized by an interval [a,b] and , where , then a vector field X along \gamma (and in particular, the value of this vector field at ) is called the parallel transport of \xi along \gamma if • \nabla_{\gamma'(t)}X=0, \text{ for all } t\in [a,b] • X_{\gamma(a)}=\xi. Formally, the first condition means that X is parallel with respect to the pullback connection on the pullback bundle . However, in a local trivialization it is a first-order system of linear ordinary differential equations, which has a unique solution for any initial condition given by the second condition (for instance, by the Picard–Lindelöf theorem). The parallel transport of X \in T_{\gamma(s)} M to the tangent space T_{\gamma(t)} M along the curve \gamma : [0,1] \to M is denoted by . The map : \Gamma(\gamma)_s^t : T_{\gamma(s)} M \to T_{\gamma(t)} M is linear. In fact, it is an isomorphism. Let \overline\gamma : [0,1] \to M be the inverse curve \overline\gamma(t) = \gamma(1-t). Then \Gamma(\overline\gamma)_t^s is the inverse of . To summarize, parallel transport provides a way of moving tangent vectors along a curve using the affine connection to keep them "pointing in the same direction" in an intuitive sense, and this provides a linear isomorphism between the tangent spaces at the two ends of the curve. The isomorphism obtained in this way will in general depend on the choice of the curve. If it does not, then parallel transport along every curve can be used to define parallel vector fields on , which can only happen if the curvature of is zero. A linear isomorphism is determined by its action on an ordered basis or frame. Hence parallel transport can also be characterized as a way of transporting elements of the (tangent) frame bundle along a curve. In other words, the affine connection provides a lift of any curve in to a curve in . Examples The images below show parallel transport induced by the Levi-Civita connection associated to two different Riemannian metrics on the punctured plane {{tmath| \mathbf R^2 \smallsetminus \{0,0\} }}. The curve the parallel transport is done along is the unit circle. In polar coordinates, the metric on the left is the standard Euclidean metric , while the metric on the right is . This second metric has a singularity at the origin, so it does not extend past the puncture, but the first metric extends to the entire plane. Warning: This is parallel transport on the punctured plane along the unit circle, not parallel transport on the unit circle. Indeed, in the first image, the vectors fall outside of the tangent space to the unit circle. Since the first metric has zero curvature, the transport between two points along the circle could be accomplished along any other curve as well. However, the second metric has non-zero curvature, and the circle is a geodesic, so that its field of tangent vectors is parallel. Metric connection A metric connection is any connection whose parallel transport mappings preserve the Riemannian metric, that is, for any curve \gamma and any two vectors {{tmath| X, Y \in T_{\gamma(s)}M }}, : \langle\Gamma(\gamma)_s^tX,\Gamma(\gamma)_s^tY\rangle_{\gamma(t)}=\langle X,Y\rangle_{\gamma(s)}. Taking the derivative at t = 0, the operator ∇ satisfies a product rule with respect to the metric, namely : Z\langle X,Y\rangle = \langle \nabla_ZX,Y\rangle + \langle X,\nabla_Z Y\rangle. Relationship to geodesics An affine connection distinguishes a class of curves called (affine) geodesics. A curve \gamma:I\rightarrow M is an affine geodesic if \dot\gamma is parallel transported along , 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. If \nabla is a metric connection, then the affine geodesics are the usual geodesics of Riemannian geometry and are the locally distance minimizing curves. More precisely, first note that if \gamma: I\rightarrow M, where I is an open interval, is a geodesic, then the norm of \dot\gamma is constant on . Indeed, :\frac{d}{dt}\langle\dot\gamma(t),\dot\gamma(t)\rangle = 2\langle\nabla_{\dot\gamma(t)}\dot\gamma(t),\dot\gamma(t)\rangle =0. It follows from an application of Gauss's lemma that if A is the norm of \dot\gamma(t) then the distance, induced by the metric, between two close enough points on the curve , say \gamma(t_1) and , is given by \mbox{dist}\big(\gamma(t_1),\gamma(t_2)\big) = A|t_1 - t_2|. The formula above might not be true for points which are not close enough since the geodesic might for example wrap around the manifold (e.g. on a sphere). == Parallel transport on a vector bundle ==

Parallel transport of tangent vectors is a special case of a more general construction involving an arbitrary vector bundle E. Specifically, parallel transport of tangent vectors is the case where E is the tangent bundle TM. Let M be a smooth manifold. Let E → M be a vector bundle with connection ∇ and γ: I → M a curve parameterized by an open interval I. A section X of E along γ is called parallel if : \nabla_{\dot\gamma(t)}X=0\text{ for }t \in I. In the case when E is the tangent bundle whereby X is a tangent vector field, this expression means that, for every t in the interval, tangent vectors in X are "constant" (the derivative vanishes) when an infinitesimal displacement from \gamma(t) in the direction of the tangent vector \dot{\gamma}(t) is done. Suppose we are given an element e0 ∈ EP at P = γ(0) ∈ M, rather than a section. The parallel transport of e0 along γ is the extension of e0 to a parallel section X on γ. More precisely, X is the unique part of E along γ such that • \nabla_{\dot\gamma} X = 0 • X_{\gamma(0)} = e_0. Note that in any given coordinate patch, (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. Thus the connection ∇ 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 γ(s) to that over γ(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 M. This is only possible if the curvature of ∇ 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 ∇ at x. There is a close relation between this group and the value of the curvature of ∇ at x; this is the content of the Ambrose–Singer holonomy theorem. Recovering the connection from the parallel transport Given a covariant derivative ∇, the parallel transport along a curve γ is obtained by integrating the condition {{tmath|1= \textstyle{\nabla_{\dot{\gamma} }=0} }}. Conversely, if a suitable notion of parallel transport is available, then a corresponding connection can be obtained by differentiation. This approach is due, essentially, to ; see . also adopts this approach. Consider an assignment to each curve γ in the manifold a collection of mappings : \Gamma(\gamma)_s^t : E_{\gamma(s)} \rightarrow E_{\gamma(t)} such that • \Gamma(\gamma)_s^s = Id, the identity transformation of Eγ(s). • \Gamma(\gamma)_u^t\circ\Gamma(\gamma)_s^u = \Gamma(\gamma)_s^t. • The dependence of Γ on γ, s, and t is "smooth". 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. Nevertheless, given such a rule for parallel transport, it is possible to recover the associated infinitesimal connection in E as follows. Let γ be a differentiable curve in M with initial point γ(0) and initial tangent vector X = γ′(0). If V is a section of E over γ, then let : \nabla_X V = \lim_{h\to 0}\frac{\Gamma(\gamma)_h^0V_{\gamma(h)} - V_{\gamma(0)}}{h} = \left.\frac{d}{dt}\Gamma(\gamma)_t^0V_{\gamma(t)}\right|_{t=0}. This defines the associated infinitesimal connection ∇ on E. One recovers the same parallel transport Γ from this infinitesimal connection. == Generalizations ==

The parallel transport can be defined in greater generality for other types of connections, not just those defined in a vector bundle. One generalization is for principal connections . Let P → M be a principal bundle over a manifold M with structure Lie group G and a principal connection ω. As in the case of vector bundles, a principal connection ω on P defines, for each curve γ in M, a mapping : \Gamma(\gamma)_s^t : P_{\gamma(s)} \rightarrow P_{\gamma(t)} from the fibre over γ(s) to that over γ(t), which is an isomorphism of homogeneous spaces: i.e. \Gamma_{\gamma(s)} g = g\Gamma_{\gamma(s)} for each . Further generalizations of parallel transport are also possible. In the context of Ehresmann connections, where the connection depends on a special notion of "horizontal lifting" of tangent spaces, one can define parallel transport via horizontal lifts. Cartan connections are Ehresmann connections with additional structure which allows the parallel transport to be thought of as a map "rolling" a certain model space along a curve in the manifold. This rolling is called development. == Approximation: Schild's ladder ==

. The segments A1X1 and A2X2 are an approximation to first order of the parallel transport of A0X0 along the curve. Parallel transport can be discretely approximated by Schild's ladder, which takes finite steps along a curve, and approximates Levi-Civita parallelogramoids by approximate parallelograms. == See also ==