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1. The text stresses that in the absence of something like covariant derivative, we have no way of connecting tangent spaces of different points with each other. What confuses me is how can we have a notion of a smooth vector field in that case? I understand the definition of a smooth vector field (tangent vectors acting as derivations turn smooth functions to smooth functions); since we have it, doesn't it mean that we do have a notion of what it means for tangent vectors from two nearby tangent spaces to be very close?

Also, why doesn't the standard trick of passing to R^n work? Given two nearby points on the manifold, why can't we translate them into R^n by a chart and identify their tangent spaces in R^n? If this leads to different results in different charts, why wouldn't the smoothness of chart transformations make everything alright, and is there an instructive example to show this failure?

2. Is there a good way of understanding tensor product and contraction geometrically? I can follow all the indices around, I just completely fail to understand what it *means*, and it's very frustrating. For example, Wald defines parallel transport of vector [tex]v^{b}[/tex] along a curve with tangent [tex]t^{a}[/tex] by the equation [tex]t^{a}\nabla_{a}v^{b} = 0[/tex]. But I don't understand

*why*this should capture the notion of parallel transport. [tex]\nabla_{a}v^{b}[/tex] is some (1,1)-tensor, and while I can write out the definition of what it means to multiply it by [tex]t^{a}[/tex] and contract over index a, I don't really understand what it means. Even worse, for a general tensor T with both covariant and contravariant indices, I have no clue how to imagine [tex]t^{a}\nabla_{a}T = 0[/tex]. Is there a helpful way to visualize/understand this?

Many thanks in advance!