Which book about differential geometry will have these formula about torsion tensor?
$$\nabla_{j}T^{i}_{kl}+\nabla_{k}T^{i}_{lj}+\nabla_{l}T^{i}_{jk}=R^{i}_{jkl}+R^{i}_{klj}+R^{i}_{ljk}$$
$$\left[\nabla_i, \nabla_j \right]X^k = R^k_{lij}X^l+T^l_{ij}\nabla_l X^k $$
I read many books about differential geometry, but all treat the torsion-free connection. Which book will discuss the torsion and affine connection in detail and have the formula about Bianchi's identity with torsion? Thanks!
The second formula, but with the extra minus sign: $$\left[\nabla_i, \nabla_j \right]X^k = -R^k_{lij}X^l+T^l_{ij}\nabla_l X^k $$ can be found in http://www.worldscientific.com/worldscibooks/10.1142/3812 (S.S. Chern, W.H. Chen and K.S. Lam, Lectures on Differential Geometry) formula 2.46 on page 121. The first identity, I think, is a simple consequence of definitions of R and T tensors through the connection. How the standard geometrical definitions and formulas are changed under the presence of torsion is explained in http://www.slimy.com/~steuard/teaching/tutorials/GRtorsion.pdf (General Relativity with Torsion: Extending Wald’s Chapter on Curvature, by Steuard Jensen).
