Parallel Transport
Let \((M,g)\) be a smooth Riemannian manifold. Write \(\nabla\) for its Levi–Civita connection and \(T_pM\) for the tangent space at \(p\). The coordinate conventions are those in tangent spaces and derivations.
A connection gives a way to keep a tangent vector parallel while its base point moves along a curve. The resulting transport compares the tangent spaces at the endpoints. Although the transport preserves inner products for the Levi–Civita connection, it can depend on the path taken. Again, details related to flows on vector fields can be found in Appendix Vector Fields and Their Flows.
Definition 1. Vector field along curve \(c\), denoted \(X\), where \(c : I \to M\) and \(I \subset \mathbb R\), is a map \(X: I \to TM\) such that \(X(t) \in T_{c(t)} M\) for all \(t \in I\).
Theorem 2. Let \(c:I\to M\) be a smooth curve and \(\nabla\) a smooth connection. There is a unique real-linear covariant derivative \(D_t\) on smooth vector fields along \(c\) such that
and, whenever \(Y(t)=X(c(t))\) for a smooth field \(X\) on a neighbourhood of the curve, \(D_tY=\nabla_{\dot c}X\). The same construction applies on each smooth piece of a piecewise smooth curve.
Proof. In local coordinates write \(Y(t)=Y^j(t)\partial_j|_{c(t)}\). The required rules force
Indeed each coordinate basis field extends to the chart, so the extension rule fixes its derivative; the product rule then fixes the derivative of every linear combination. This proves local uniqueness even for fields along a self-intersecting curve, which need not extend to a single ambient field.
Conversely, the displayed formula is linear and satisfies the product rule. If \(Y=X\circ c\), the chain rule gives \(\dot Y^k=(\partial_iX^k)(c)\dot c^i\), so the formula is exactly \(\nabla_{\dot c}X\). Formulas from two charts agree on their overlap: express one coordinate frame as a smooth linear combination of the other, apply the product rule, and use the connection transformation law for the derivatives of those extending frame fields. Equivalently, uniqueness for the forced expression in any local frame identifies them. Thus the local operators patch to a global operator along \(c\). ◻
Definition 3. A vector field \(X\) along a curve \(c\) is parallel along \(c\) if \(\frac{\Dif X}{\dif t} = 0\).
Definition 4 (Parallel transport). Fix \(t_0\in I\). For \(v\in T_{c(t_0)}M\), let \(X_v\) be the unique parallel field with \(X_v(t_0)=v\). Parallel transport to time \(t\) is
When \(t_0=0\), we abbreviate this to \(P_t\). It maps an initial vector to a vector in the endpoint fibre, not to the entire field.
Existence and uniqueness follow from the linear system
On every compact subinterval covered by one chart its coefficients are bounded; the estimate \(|X(t)|\leq |X(t_0)|\exp(\int\|A(s)\|ds)\) rules out finite-time blow-up. A finite chart subdivision covers any compact portion of the curve, and uniqueness patches the solutions. The resulting field exists at every time of \(I\). Linearity of the equation makes \(P_{t\leftarrow t_0}\) linear. Uniqueness gives \(P_{u\leftarrow t}P_{t\leftarrow s}=P_{u\leftarrow s}\), so reversed transport is its inverse.
Proposition 5. The parallel transport defines an isometry from \(T_{c(0)} M\) to \(T_{c(t)} M\) for all \(t\). More generally, if \(X\) and \(Y\) are vector fields along curve \(c\), then
Proof. Expand \(\langle X,Y\rangle=g_{ij}(c)X^iY^j\) and differentiate. The terms from \(\dot X\) and \(\dot Y\) are immediate. Metric compatibility,
rewrites the remaining metric-derivative term as the two connection terms in the stated identity. If \(X,Y\) are parallel, both right-hand terms vanish, so their inner product is constant. Transport is already a linear isomorphism by the preceding ODE argument; preservation of all inner products makes it an isometry. ◻
Example 6 (Transport around a latitude). On the round sphere of radius \(R\), let \(\theta\) denote colatitude, measured from the north pole, and let \(\varphi\) denote longitude. Fix \(0<\theta_0<\pi\) and traverse the circle \(\theta=\theta_0\) with increasing \(\varphi\) from \(0\) to \(2\pi\). The metric is \(R^2(d\theta^2+\sin^2\theta\,d\varphi^2)\). Its relevant Christoffel symbols are
These follow from \(\partial_\theta g_{\varphi\varphi}=2R^2\sin\theta\cos\theta\); all other derivatives needed here vanish. Taking longitude as the curve parameter, parallel transport is therefore
Use the orthonormal frame \(e_\theta=R^{-1}\partial_\theta\), \(e_\varphi=(R\sin\theta_0)^{-1}\partial_\varphi\). Writing \(V=a e_\theta+b e_\varphi\), or \(a=RV^\theta\), \(b=R\sin\theta_0V^\varphi\), gives
Consequently
The original first-order system couples the integration constants. For arbitrary initial components \((a_0,b_0)\), its solution is
In particular a unit initial vector \(e_\theta\) gives
Its squared norm is \(a^2+b^2=1\), as parallel transport requires. The coordinate frame agrees at longitudes zero and \(2\pi\), so the signed rotation in the ordered frame \((e_\theta,e_\varphi)\) is \(-2\pi\cos\theta_0\) modulo \(2\pi\). If latitude is \(\lambda=\pi/2-\theta_0\), this is \(-2\pi\sin\lambda\). A cosine comparison determines only an unsigned angle modulo sign; the first-order rotation matrix supplies the sign. At the equator the transport returns without rotation. The poles are outside these latitude coordinates, but the rotation tends to zero modulo \(2\pi\) as the loop shrinks to a pole. This is the geometric parallel-transport contribution to the idealized Foucault precession.
Remark 7. It is common to use \(\nabla\) and \(\frac{\Dif}{\dif t}\) interchangeably. For statements of results that involve differentiable (Riemannian) manifolds, \(\nabla\) will always mean the Levi-Civita connection, unless otherwise stated. The introduction of the notation \(\frac{\Dif}{\dif t}\) is to make clear the construction of this (uniquely defined!) operator.
The final application concerns geodesic symmetries. To ask whether reflecting geodesics through a point preserves the metric, we first define an isometry.
Definition 8. Let \(M\), \(N\) be differentiable manifolds. Let \(h : M \to N\) be a diffeomorphism. \(h\) is an isometry if for all \(p \in M\) and \(v,w \in T_p M\),
where \(\langle \cdot,\cdot \rangle_M\) and \(\langle \cdot,\cdot \rangle_N\) denote the Riemannian metrics on \(M\) and \(N\), respectively.
Definition 9. A smooth map \(h:M\to N\) is a local isometry if every \(p\in M\) has an open neighbourhood \(U\) such that \(h|_U\) is a diffeomorphism onto an open subset of \(N\) and preserves the metric there. The map need not be globally one-to-one.
Remark 10 (Locally symmetric spaces). Choose a normal neighbourhood \(U=\exp_p(B_\varepsilon(0))\). The geodesic symmetry at its centre is
It fixes \(p\), squares to the identity, and has differential \((ds_p)_p=-I\). A manifold is locally symmetric when each such map is an isometry on a sufficiently small neighbourhood. Equivalently, each point admits a local isometry fixing it with differential \(-I\): an isometry preserves the Levi–Civita connection by the Koszul uniqueness theorem, hence takes the geodesic with initial data \((p,v)\) to the one with data \((p,-v)\). A symmetric space has such symmetries as global isometries at every point.
We prove the equivalent curvature criterion \(\nabla R=0\). An isometry preserves the connection, its curvature, and the covariant derivative of curvature. Thus, at a fixed point of a symmetry,
The left side is the negative of the original vector, whereas the four input signs on the right multiply to plus. The tensor must therefore be zero. Since every point is a symmetry centre, \(\nabla R=0\) everywhere.
Conversely assume \(\nabla R=0\), and fix \(v,w\in T_pM\) with \(v\) small. Let \(\gamma_v(t)=\exp_p(tv)\) for \(0\leq t\leq1\), and let \(P_v(t)\) be parallel transport along it. The variation \(\exp_p(t(v+sw))\) has variational field \(J_v\) satisfying
This equation follows by differentiating the geodesic equation and using Lemma 26 in the connections note, with the curvature convention used there. Its endpoint is \(J_v(1)=(d\exp_p)_v w\).
Write \(j_v(t)=P_v(t)^{-1}J_v(t)\). Parallelness of \(R\) and of \(\dot\gamma_v\) turns the equation into the constant-coefficient vector equation
For \(-v\) with initial variation \(-w\), the coefficient is unchanged, since \(R_p(-v,z)(-v)=R_p(v,z)v\). Uniqueness of this linear ODE gives \(j_{-v,-w}=-j_{v,w}\). Parallel transport preserves inner products, so, for any \(w_1,w_2\),
Differentiating \(s_p\circ\exp_p(v)=\exp_p(-v)\) shows this is exactly metric preservation by \(ds_p\). Since \(\exp_p\) is a diffeomorphism on the normal ball, \(s_p\) is a local isometry. This proves the converse directly; the later Cartan local-isometry theorem provides a more general version of the same Jacobi-field comparison. No globally unique minimizing geodesic between arbitrary points was assumed.
Reading: Carmo (1992), Riemannian geometry.