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I have been reading section 3.1 of Wald's GR book in which he introduces the notion of a covariant derivative. As I understand, this is introduced as the (partial) derivative operators $\partial_{a}$ are dependent on the coordinate system one chooses and thus not naturally associated with the structure of the manifold, therefore we introduce a new derivative operator $\nabla_{a}$ which fulfils this criteria. Intuitively, the covariant derivative is constructed in such a way as to remove any artefacts arising from any particular coordinate system, such that the rate of change of a tensor field in a direction along the manifold is itself a tensor field. Is this correct?

He then goes on to say that a vector $\mathbf{v}^{a}$ given at each point along a curve $C$ (with tangent vector $\mathbf{t}^{a}$) is said to be parallely transported as one moves along the curve if the equation $$\mathbf{t}^{a}\nabla_{a}\mathbf{v}^{b}=0$$ is satisfied along the curve. Intuitively, is this a statement that a vector $\mathbf{v}^{a}$ is parallely transported along the curve if it remains constant as it moves along the curve, this can be translated into the mathematical statement that its directional derivative along the curve vanishes, i.e. $\mathbf{t}^{a}\nabla_{a}\mathbf{v}^{b}=0$. Is this correct?

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    $\begingroup$ Hint: Questions which can be answered with a simple Yes aren't good questions. $\endgroup$ – ACuriousMind Feb 20 '15 at 13:22
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    $\begingroup$ Apologies, I'm just trying to get an understanding of the notion really and thought it might be useful to express my thoughts on the matter so far. $\endgroup$ – Will Feb 20 '15 at 13:45
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In a sense, parallel transport, covariant derivative and connection are all synonym for you can recover one from the other. So given a manifold one usually starts by giving one notion (e.g. how a vector field is transported parallel to itself along a family of curves) and then, if needed, the other objects are derived. In physics, when dealing with a (pseudo-)Riemannian manifold, one makes use of the Levi-Civita, or torsion-free metric connection, which in terms of coordinates is completely specified by the Christoffel symbols (which are not tensor quantities alone).

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