I'm having some trouble solving the following problem, could you please give me a solution? I know what covariant derivative is, but no idea about the manifold they mention and what particular tensor they're talking about: it's from an exam in relativity and it seems to use different langiages than mathematicians are normally used to: a compete solution would be great!
a) Consider a congruence of curves $x^{a}= x^{a}(u)$ defined in such way that there is only one curve through each point of the manifold. Using this congruence, introduce a vector field over the whole manifold. Find the Lie derivative with respect to this vector field of the contravariant components $T^{ab}$ of the rank two tensor field $T.$
b) The covariant derivative of the contravariant component $X^{a}$ of the vector field $X$ is given by $\nabla_{c}X^{a}= \partial_{c}X^{a} + \Gamma_{bc}^{a}X^{b}.$
and the covariant derivative of the mixed components of the rank 2 tensor field $T$ is given by:
$$\nabla_{d}T^{a}_c = \partial_{d}T^{a}_c + \Gamma^{a}_{ed}T^{e}_c- \Gamma^{e}_{cd}T^{a}_e$$
Using the commutator of the contravariant vector field $Y^{a}$ derive the expression for the Riemann curvature tensor.