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## Main Question or Discussion Point

Is there any relationship between the Lie ([itex]\pounds[/itex]) and covariant derivative ([itex]\nabla[/itex])?

Say I have 2 vector fields V, W and a metric g, the Lie and covariant derivative of W along V are:

[tex]\pounds_{V}W = [V,W][/tex]

[tex]V^\alpha \nabla_\alpha W^\mu = V^\alpha \partial_\alpha W^\mu + V^\alpha \Gamma^\mu_{\alpha \nu} W^\nu[/tex]

which appear rather different.

But conceptually I thought both derivatives help us to define what parallel transporting a vector in a general manifold means? Is there a good place to read about such issues?

Thanks!!

Say I have 2 vector fields V, W and a metric g, the Lie and covariant derivative of W along V are:

[tex]\pounds_{V}W = [V,W][/tex]

[tex]V^\alpha \nabla_\alpha W^\mu = V^\alpha \partial_\alpha W^\mu + V^\alpha \Gamma^\mu_{\alpha \nu} W^\nu[/tex]

which appear rather different.

But conceptually I thought both derivatives help us to define what parallel transporting a vector in a general manifold means? Is there a good place to read about such issues?

Thanks!!