Curvature

Covariant derivatives commute on scalars, but not on other tensors. So, consider $[{\nabla }_{\mu },{\nabla }_{\nu }]V^{\lambda }={\nabla }_{\mu }({\nabla }_{\nu }{V}^{\lambda })-{\nabla }_{\nu }({\nabla }_{\mu }{V}^{\lambda })$ Recall ${\nabla }_{\mu }{T}_{\nu }^{\lambda }={\partial }_{\mu }{T}_{\nu }^{\lambda }+{\Gamma }_{\mu \rho }^{\lambda }{T}_{\nu }^{\rho }-{\Gamma }_{\mu \nu }^{\rho }{T}_{\rho }^{\lambda }$ Then, $[{\nabla }_{\mu },{\nabla }_{\nu }]V^{\lambda }=[{\partial }_{\mu }{\Gamma }_{\nu \rho }^{\lambda }-{\partial }_{\nu }{\Gamma }_{\mu \rho }^{\lambda }+{\Gamma }_{\mu \sigma }^{\lambda }{\Gamma }_{\nu \rho }^{\sigma }-{\Gamma }_{\nu \sigma }^{\lambda }{\Gamma }_{\mu \rho }^{\sigma }]V^{\rho }$ We define a (1,3) type tensor called theRiemann-Tensor or the Curvature Tensor $R_{\rho \mu \nu }^{\lambda }={\partial }_{\mu }{\Gamma }_{\nu \rho }^{\lambda }-{\partial }_{\nu }{\Gamma }_{\mu \rho }^{\lambda }+{\Gamma }_{\mu \sigma }^{\lambda }{\Gamma }_{\nu \rho }^{\sigma }-{\Gamma }_{\nu \sigma }^{\lambda }{\Gamma }_{\mu \rho }^{\sigma }$ and hence $[{\nabla }_{\mu },{\nabla }_{\nu }]V^{\lambda }=R_{\rho \mu \nu }^{\lambda }{V}^{\rho }$

• The commutator is always related to theRiemann-Tensor . For a one-form, $[{\nabla }_{\mu },{\nabla }_{\nu }]{\omega }_{\lambda }=-R_{\rho \mu \nu }^{\lambda }{\omega }_{\rho }$
• If spacetime is flat (i.e., the metric isMinkowski ), then theChristoffel-symbols vanish and henceRiemann-Tensor is also zero. The converse is also true. Hence, spacetime is flat iffRiemann-Tensor is zero.

1. Path Independence of parallel transport
2. Geodesic Deviation
3. Riemann Normal Coordinates
4. Symmetries and Ricci Tensor