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.

Transport defining a nonrotating frame along an accelerated timelike curve. With four-velocity normalization $u^{\mu }{u}_{\mu }=-c^2$ and four-acceleration $a^{\mu }=D{u}^{\mu }/d\tau$, the displayed equation preserves inner products and keeps a spatial spin orthogonal to $u$. Along a geodesic, $a^{\mu }=0$ and it reduces to parallel transport.

A vector field along a smooth curve is parallel if its covariant derivative along that curve is zero. Its value is the parallel transport of its initial value. A Levi-Civita connection preserves inner products under this transport, so the length of a parallel field is constant. Along a unit-speed curvature-one geodesic, the normal Jacobi field with zero initial value and initial derivative $a$ is $\sin t$ times the parallel transport of $a$.

For the positively oriented boundary of a spherical triangle on the unit sphere, parallel transport is rotation by its angle excess modulo $2\pi$. The tangent is parallel along each geodesic side and turns by $\pi -\alpha$ at a corner of angle $\alpha$. The three corner turns sum to $3\pi -\alpha -\beta -\gamma$, and combining them with parallel transport returns the initial tangent. Hence the transport angle is the negative of that sum modulo $2\pi$, giving the formula. The spherical excess formula identifies the angle with enclosed area; reversing traversal changes its sign.

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