= Solution
Ordinary component differences at neighboring points depend on how the local bases vary. The <covariant derivative> corrects for that variation:
$$
\boxed{\nabla_b V^a=\partial_bV^a+\Gamma^a_{bc}V^c,
\qquad \nabla_bW_a=\partial_bW_a-\Gamma^c_{ba}W_c.}
$$
Each upper <tensor> index adds a connection term and each lower index subtracts one. A <scalar> has $\nabla_a f=\partial_a f$, while the derivative of a <tensor> transforms as a <tensor> with one additional covariant index. The <Christoffel symbols> themselves are not <tensor> components; their inhomogeneous transformation cancels that of the ordinary derivative.
For the <Levi-Civita connection>,
$$
\Gamma^a_{bc}=\frac12g^{ad}
(\partial_b g_{dc}+\partial_c g_{bd}-\partial_d g_{bc}),
\qquad \nabla_cg_{ab}=0,\qquad\Gamma^a_{bc}=\Gamma^a_{cb}.
$$
These express <metric compatibility> and zero torsion. Along $x^a(\lambda)$, <parallel transport> is the <initial value problem>
$$
\boxed{\frac{DV^a}{d\lambda}
=\frac{dV^a}{d\lambda}
+\Gamma^a_{bc}\frac{dx^b}{d\lambda}V^c=0.}
$$
It preserves <inner products> for the <Levi-Civita connection>. <Parallel transport> of the tangent itself gives the affinely parametrized <geodesic equation>, $\ddot x^a+\Gamma^a_{bc}\dot x^b\dot x^c=0$.
Transport generally depends on the path. On a round <sphere>, transport a <tangent vector> around the triangle bounded by two meridians $90$ degrees apart and the intervening equatorial arc. All three sides are <geodesics>; the enclosed spherical area is $\pi a^2/2$. The returning <vector> is rotated by $90$ degrees, with sign set by traversal orientation. Locally this <holonomy> is controlled by <curvature>, even though the connection can be made zero at a single point. A <vector> can therefore have changing coordinate components while remaining parallel, and constant components do not generally mean <parallel transport>.
\Image[/past-exam-of-the-relativity-course-of-the-university-of-cambridge/2015/paper-1-spherical-transport.png]
{title=Original spherical-triangle transport: a tangent initially pointing along the first meridian returns rotated by ninety degrees}
This construction compares directions at separate points through a chosen connection, rather than assuming that a curved manifold supplies a path-independent comparison.<Lecture Notes on General Relativity by Sean M. Carroll>[Section 3, Curvature: the covariant derivative, Levi-Civita connection and parallel transport]{ref}
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