Parallel frame along a curve ($D_t{E}_a=0$)

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.