The Parallel Transport Operator

Introduction

A connection on a Riemannian manifold answers the question of how to compare tangent vectors at different points, and the answer is the parallel transport: along a curve, a vector is moved by the rule that its covariant derivative vanishes. For each curve the transport is an invertible linear map between the tangent spaces at its ends, and because the Levi-Civita connection is metric, the map is a linear isometry. The article reads the transport as an operator: it is the operator that moves the tangent space along the curve, it composes along curves, and the loop transports form the holonomy group, the operator invariant of the curvature that the rest of the theory — the decomposition of a manifold into its irreducible pieces, the parallel tensors, the classification of the possible geometries — is built on.

The article develops the transport operator. It defines the parallel transport along a curve and proves that it is a linear isometry, that it composes, and that its inverse is the transport along the reversed curve; it shows that the transport extends to the tensor bundles and that the holonomy group is the group of the loop transports; it identifies the connection with the derivative of the transport, so that the curvature is the second derivative of the transport around an infinitesimal loop; it states and proves the holonomy principle, that a tensor field is parallel exactly when it is fixed by the holonomy; and it states the theorem of Ambrose–Singer, that the Lie algebra of the holonomy group is generated by the curvature operators transported along the paths. The examples are the flat spaces, the spheres and the surfaces of revolution.

The article assumes the metric, the Levi-Civita connection, the geodesics and the curvature of Curvature and Geodesics and Riemannian Geometry, and the curve and the vector field along a curve of Smooth Manifolds and Differential Geometry. The general connection on a vector bundle and the transport in a bundle are Fibre Bundles, Connections and Curvature of Part III; the classification of the holonomy groups, the Berger list, the de Rham decomposition and the holonomy representation are The Holonomy and the Curvature, later in this category. The transport operator is the one whose adjoint and involution appear in the - * Operator Theory articles of the group. No physics is invoked.

The Transport Operator Along a Curve

Definition and First Properties

Definition. Let $\gamma : [a, b] \to M$ be a piecewise smooth curve in a Riemannian manifold $(M, g)$ with Levi-Civita connection $\nabla$. A vector field $V$ along $\gamma$ is parallel if its covariant derivative along the curve vanishes,

$$ \frac{DV}{dt} = \nabla_{\gamma'(t)}V(t) = 0 . $$

For the parallel fields, which form an $n$-dimensional space because the equation is a first-order linear system, the evaluation $V(a) \mapsto V(b)$ is a linear isomorphism

$$ P_\gamma : T_{\gamma(a)}M \longrightarrow T_{\gamma(b)}M, $$

the parallel transport along $\gamma$, and it is characterised by $P_\gamma(v) = V(b)$ where $V$ is the parallel field with $V(a) = v$, and $\frac{D}{dt}(P_t v) = 0$ for the transport $P_t$ along $\gamma|_{[a,t]}$.

Theorem. The transport along $\gamma$ is a linear isometry,

$$ g_{\gamma(b)}\bigl(P_\gamma(v), P_\gamma(w)\bigr) = g_{\gamma(a)}(v, w), $$

it depends only on the curve and not on its parametrisation up to reparametrisation, it is the identity along a constant curve, and the transports compose and reverse:

$$ P_{\gamma_2 * \gamma_1} = P_{\gamma_2}\circ P_{\gamma_1}, \qquad P_{\gamma^{-1}} = (P_\gamma)^{-1} = (P_\gamma)^{*}, $$

where $\gamma_2 * \gamma_1$ is the concatenation, $\gamma^{-1}$ is the reversed curve, and the star is the adjoint for the two metrics; consequently the transport along a closed curve starting and ending at $p$ is an element of the orthogonal group $O(T_pM, g_p)$. The transport depends only on the homotopy class of the curve when the curvature vanishes on the enclosed region.

Proof. For two parallel fields $V, W$ along $\gamma$ one has $\frac{d}{dt}g(V,W) = g(\frac{DV}{dt},W)+g(V,\frac{DW}{dt}) = 0$ by the metricity of $\nabla$, so $g(V,W)$ is constant and the transport is an isometry. A reparametrisation multiplies $\gamma'$ by a scalar and the covariant derivative by the same scalar, so the equation $DV/dt = 0$ is unchanged. The transport along a constant curve fixes every vector, and the covariant derivative along a concatenation is the covariant derivative along each piece, which gives the composition; the reversed parallel field along $\gamma^{-1}$ has energy of the same sign and gives the inverse, which is also the adjoint by the isometry. The homotopy statement is the holonomy theorem below in the flat case, which is the classical monodromy of a flat connection and is Fibre Bundles, Connections and Curvature read on the tangent bundle.

The Transport of Tensors

Proposition. The transport extends to the tensor bundles: for a tensor $T$ at $\gamma(a)$ the transported tensor $P_\gamma T$ at $\gamma(b)$ is defined by transporting each argument, and the parallel tensors along $\gamma$ are exactly the images of the tensors at the initial point. The transport preserves the metric and the volume form,

$$ P_\gamma g = g, \qquad P_\gamma\mathrm{vol}_g = \pm\,\mathrm{vol}_g, $$

the sign being $+$ exactly when the transport preserves the orientation.

Proof. On a decomposable tensor the transport is defined by applying $P_\gamma$ to each factor, and the definition extends by linearity; the parallel tensors are the fixed points of the equation $\frac{D}{dt}=0$, whose solutions are the transported tensors. The metric is a parallel tensor because the connection is metric; the volume form is parallel because it is the metric contraction of the orientation, and a metric-preserving isomorphism of $T_pM$ carries an orthonormal frame to an orthonormal frame.

The Holonomy Group

The Group of Loop Transports

Definition. Fix $p \in M$. The holonomy group at $p$ is the subgroup of $O(T_pM, g_p)$ of the loop transports,

$$ \operatorname{Hol}_p = \bigl\{P_\gamma : \gamma \text{ a piecewise smooth closed curve based at } p\bigr\}, $$

and the restricted holonomy group $\operatorname{Hol}^0_p$ is the subgroup of the transports along the null-homotopic loops. Both are subgroups of the orthogonal group, $\operatorname{Hol}^0_p$ is the identity component of $\operatorname{Hol}_p$, and the groups at different points are conjugate:

$$ \operatorname{Hol}_q = P_\gamma\,\operatorname{Hol}_p\,P_\gamma^{-1}, \qquad q = \gamma(1). $$

Proof. The loop transports form a subgroup because the concatenation of two loops is a loop, the reverse of a loop is a loop, and the transport along a constant loop is the identity; the isometry is the first theorem above. The restricted group is normal in $\operatorname{Hol}_p$: for a loop $\eta$ null-homotopic at $p$ and a path $\gamma$ from $p$ to $p$, the conjugate loop $\gamma^{-1}*\eta*\gamma$ is null-homotopic, so $P_\gamma^{-1}P_\eta P_\gamma = P_{\gamma^{-1}*\eta*\gamma}\in\operatorname{Hol}^0_p$. The restricted group is the identity component of $\operatorname{Hol}_p$: it is a subgroup of the identity component because a null-homotopic loop can be contracted through the loops based at $p$, giving a continuous path of the transports to the identity, and it contains the identity component because the holonomy group is a Lie subgroup of the orthogonal group, whose identity component is connected; this is the standard theorem, proved with the closed-subgroup theorem for the holonomy group. The conjugation statement is the composition of the transports along a path from $p$ to $q$, its reverse and a loop at $p$, which is a loop at $q$ in the right class, and conversely.

The Holonomy Representation and the Reduction

Definition. The holonomy representation is the representation of the holonomy group on the tangent space, $\rho_p : \operatorname{Hol}_p \to O(T_pM, g_p)$, which is the inclusion; the transport along a path and the displacement of a frame identify the frame bundle with $\operatorname{Hol}_p$ as a principal bundle, the holonomy bundle, whose structure group is the holonomy group. A reduction of the frame bundle to a subgroup $G \subseteq O(n)$ is a $G$-structure, and the holonomy group is the smallest subgroup to which the frame bundle of $(M, g)$ reduces.

Proposition. A tensor field on $M$ is parallel, $\nabla T = 0$, if and only if it is invariant under parallel transport, and this holds if and only if its value at $p$ is fixed by the holonomy representation for every $p$; the parallel tensor fields are therefore the tensors at $p$ invariant under $\operatorname{Hol}_p$, one for each such tensor. In particular the metric is always parallel, an orientation is parallel exactly when the holonomy lies in $\mathrm{SO}(T_pM)$, and a complex structure, a Kähler form or a parallel spinor appears as a reduction of the holonomy to the corresponding subgroup.

Proof. If $\nabla T = 0$ then $T$ is parallel along every curve, so $P_\gamma T = T$ and the value at $p$ is fixed, with the argument transported. Conversely if the value at $p$ is fixed by every loop transport then the field is unchanged by transport along loops, and following the field by the exponential of a small geodesic shows $\nabla T = 0$; the correspondence is the holonomy principle. The reductions are the form in which the special geometries of Kähler Geometry and Quaternionic Geometry, later in this Part, appear.

The Connection as the Derivative of the Transport

Theorem. The covariant derivative is the derivative of the transport:

$$ \nabla_XT\big|_p = \lim_{t\to0}\frac{1}{t}\Bigl(P_{\gamma_t}^{-1}\bigl(T(\gamma(t))\bigr) - T(p)\Bigr), $$

where $\gamma_t$ is the geodesic with $\gamma'(0) = X$ and $P_{\gamma_t}$ is the transport along it; equivalently, the connection is the family of the transports, and the curvature is the transport around an infinitesimal parallelogram. For the curvature,

$$ R(X, Y)v = -\lim_{s,t\to0}\frac{1}{st}\Bigl(P^{-1}_{\partial[0,s]\times[0,t]}(v) - v\Bigr) $$

up to the ordering of the transport, where the loop is the boundary of a small coordinate parallelogram spanned by $X$ and $Y$.

Proof. The transport along the geodesic satisfies the ordinary differential equation $\frac{D}{dt}V = 0$ with $V(0) = v$, which is $\dot V(t) = -\Gamma(\gamma',V)$; substituting and taking the limit gives the covariant derivative. The parallelogram statement is the same computation one order further: the transport around the closed parallelogram differs from the identity by the curvature, which is the infinitesimal holonomy, $R(X,Y) = [\nabla_X,\nabla_Y]-\nabla_{[X,Y]}$.

Transport Along a Geodesic

Proposition. Along a geodesic $\gamma$ the transport $P_t : T_{\gamma(0)}M \to T_{\gamma(t)}M$ is the derivative of the geodesic flow at $(\gamma(0),\gamma'(0))$ read on the vertical directions, it preserves the velocity, $P_t(\gamma'(0)) = \gamma'(t)$, and the Jacobi fields along $\gamma$ are the fields $t \mapsto P_t J_0(t)$ with $J_0$ the solution of the transported Jacobi equation at the origin. The transport along $\gamma$ is the fundamental matrix of the linear system

$$ \frac{D}{dt}V = 0, \qquad V(0) = v, $$

equivalently $\dot V^k(t) = -\sum_i\Gamma^k_{ij}(\gamma(t))\gamma'^i(t)V^j(t)$.

Proof. A vector field $t\mapsto\gamma'(t)/|\gamma'|$ is parallel along a geodesic because the geodesic equation is $\frac{D}{dt}\gamma'=0$, so the velocity is preserved. The Jacobi fields are the fields $P_t$ applied to the solutions of the Jacobi equation at the origin, which is the statement that the transport trivialises the tangent bundle along the geodesic and the curvature is the transported curvature; the fundamental-matrix statement is the linear system read in a chart.

Corollary (the transported curvature). Along a path $\gamma$ the transport carries the curvature to the curvature, $P_\gamma\bigl(R(X,Y)Z\bigr) = R\bigl(P_\gamma X, P_\gamma Y\bigr)P_\gamma Z$, so the curvature is a parallel tensor, $\nabla R$ is well defined, and the manifold is locally symmetric exactly when $\nabla R = 0$, the condition studied in The Geodesic Symmetry and Locally Symmetric Spaces later in this category.

Proof. The curvature is built from the connection and the Lie bracket, and the transport is an isometry that carries the connection to itself by the composition rule; hence it carries every expression in the connection to the corresponding expression. The local symmetry statement is the equivalence of $\nabla R = 0$ with the local geodesic symmetry, which is the following article of the category.

The Holonomy Algebra and the Ambrose–Singer Theorem

Theorem (Ambrose–Singer). The Lie algebra $\mathfrak{hol}_p$ of the restricted holonomy group is spanned by the curvature operators transported to $p$:

$$ \mathfrak{hol}_p = \operatorname{span}\Bigl\{P_\gamma^{-1}\,R\bigl(X, Y\bigr)\,P_\gamma \ :\ \gamma \text{ a path from } \gamma(0) \text{ to } p,\ X, Y \in T_{\gamma(0)}M \Bigr\}, $$

where $R(X,Y)$ is read as the endomorphism $Z \mapsto R(X,Y)Z$ of the tangent space and $P_\gamma$ transports endomorphisms to $p$ by conjugation. Consequently the holonomy is generated by the curvature, the restricted holonomy is trivial exactly when the metric is locally flat, and the holonomy fixes a parallel tensor exactly when the curvature does.

Proof sketch. The curvature is the infinitesimal holonomy, by the parallelogram computation above, so the Lie algebra generated by the transported curvature operators is contained in $\mathfrak{hol}_p$. For the reverse inclusion one shows that the set of the points reachable from $p$ by the parallel transport of the frames is a submanifold whose tangent space at each point is spanned by the values of the curvature, the holonomy bundle of the previous section; the structure equation of the connection then identifies the Lie algebra of the holonomy with the span of the curvature, which is the theorem of Ambrose and Singer. The flat case follows because the curvature vanishes identically exactly when the transports are locally path-independent, and the fixed-tensor statement is the holonomy principle read on the Lie algebra.

Corollary. The metric is locally symmetric if and only if $\nabla R = 0$, and then the holonomy algebra at $p$ is spanned by the operators $R(X,Y)$, no transport being needed; the irreducible pieces of the holonomy representation give the de Rham splitting of the manifold, and the classification of the possible irreducible holonomy groups is the Berger list, both of which are The Holonomy and the Curvature later in this category.

Examples

Example (the flat and the constant-curvature spaces). In Euclidean space the connection is the usual derivative, a vector is parallel exactly when it is constant, and the transport is the identification of the tangent spaces by translation; the holonomy is trivial. On the sphere $S^2$ the transport along a geodesic triangle turns a vector by the angle equal to the area of the triangle, by the Gauss–Bonnet theorem for the spherical triangle; the holonomy is the group $\mathrm{SO}(2)$ of the rotations of the tangent space, and it is transitive on the unit tangent vectors. On hyperbolic space the transport along a geodesic triangle turns a vector by the angle equal to the defect of the triangle, with the opposite sign, and again the holonomy is $\mathrm{SO}(2)$.

Example (a surface of revolution). For a surface of revolution the transport around a parallel preserves the angle to the parallel and the angular momentum $r\sin\varphi$ by the Clairaut relation, so the holonomy is generated by the rotation through the angle of the parallel transport around that circle; this is the elementary instance of the Ambrose–Singer theorem, the curvature of the surface inside the circle being the generator.

Summary

The parallel transport $P_\gamma$ along a curve $\gamma$ is the linear isomorphism the parallel fields define between the tangent spaces at the ends, and it is an isometry because the Levi-Civita connection is metric; it composes along concatenations, inverts on the reversed curve, and its adjoint is its inverse. It extends to the tensor bundles and the parallel tensors are its fixed points. The holonomy group $\operatorname{Hol}_p$ is the group of the loop transports, its identity component is the restricted holonomy of the null-homotopic loops, the groups at two points are conjugate, and the frame bundle reduces to the holonomy bundle with the holonomy group as structure group.

The holonomy principle says that a tensor field is parallel exactly when it is fixed by the holonomy: the metric is always parallel, an orientation when the holonomy lies in the special orthogonal group, and the complex, quaternionic or spin structures when the holonomy reduces to the corresponding group. The connection is the derivative of the transport, $\nabla_XT = \lim_t\frac1t(P_t^{-1}T-T)$, and the curvature is the transport around an infinitesimal parallelogram; the transport along a geodesic is the fundamental matrix of the linear system $\frac{DV}{dt}=0$ and preserves the velocity. By Ambrose–Singer the Lie algebra of the holonomy is spanned by the curvature operators transported to $p$, so the holonomy is generated by the curvature, the metric is locally flat exactly when the restricted holonomy is trivial, and the manifold is locally symmetric exactly when $\nabla R=0$. The de Rham splitting and the Berger classification of the irreducible holonomy groups are The Holonomy and the Curvature, the following article of this category.

Summary of Notation

Symbol	Meaning
$P_\gamma$, $P_t$	Parallel transport along a curve; along a subcurve
$\frac{DV}{dt} = 0$	The parallel-field equation along a curve
$P_{\gamma_2*\gamma_1} = P_{\gamma_2}P_{\gamma_1}$	Composition of transports
$P_{\gamma^{-1}} = P_\gamma^{-1} = P_\gamma^{*}$	Reversal as inverse and as metric adjoint
$\operatorname{Hol}_p$, $\operatorname{Hol}^0_p$	Holonomy group at $p$; restricted holonomy
$\operatorname{Hol}_q = P_\gamma\operatorname{Hol}_pP_\gamma^{-1}$	Conjugation of the holonomy along a path
$\rho_p$	Holonomy representation, into $O(T_pM,g_p)$
Holonomy bundle, $G$-structure	Reduction of the frame bundle to the holonomy group
Holonomy principle	Parallel tensors $\leftrightarrow$ holonomy-fixed tensors
$\nabla_XT\big\|_p = \lim_t\frac1t(P_t^{-1}T-T)$	Connection as the derivative of the transport
$R(X,Y)$	Curvature as the infinitesimal holonomy
$\mathfrak{hol}_p$	Holonomy algebra; spanned by the transported curvature
$\nabla R = 0$	Local symmetry; holonomy generated by the curvature alone