Lecture Notes on General Relativity by Sean M. Carroll (1997)

Author's graduate lecture notes, arXiv:gr-qc/9712019, 1997. Section 2 develops tangent vectors, covectors and the failure of ordinary tensor derivatives under nonlinear coordinate changes; section 3 constructs covariant differentiation, parallel transport and curvature. Metric and curvature signs must be translated when combining conventions.