Parallel transport (${\mathcal{P}}_{\gamma }^{\nabla }$)

Parallel transport along a path $\gamma$ sends an initial fiber vector to the endpoint of the unique horizontal lift through that vector. Reversing the path gives the inverse linear map.

Along a smooth curve $\gamma :I\to M$, a horizontal lift of a linear connection on a vector bundle has fibre equation $v^{\prime }=-\Gamma (\dot{\gamma })v$. On each compact time interval the coefficient matrix is bounded, and the Gronwall inequality prevents finite-time blowup. A finite cover by vector bundle trivializations and ODE uniqueness patch local solutions. Exhausting the open interval gives a unique lift on all of $I$. Its endpoint map is linear, and reversal of the path gives its inverse; no completeness of the base manifold is needed.

A parallel frame along a curve consists of a basis of vector fields $E_a$ satisfying $D_t{E}_a=0$ for its covariant derivative along a curve. Transporting a basis at the initial point constructs the frame. If the connection preserves the metric, an initially orthonormal basis stays orthonormal. In this frame the Jacobi equation becomes a linear matrix ordinary differential equation.

If $P={\mathcal{P}}_{\gamma }^{\nabla }$, the induced transport on the endomorphism bundle is conjugation:

For $d{s}^2=d{\theta }^2+{\sin }^2\theta d{\varphi }^2$, a vector initially having coordinate components $(V^{\theta },V^{\varphi })=(1,0)$ returns after one constant-$\theta$ circuit with