Levi-Civita Connection

1. Torsion-free connection
2. Metric compatible connection ${\nabla }_{\lambda }{g}_{\mu \nu }=0$ By writing the 3 equations for the covariant derivatives of the metric, we find ${\Gamma }_{\mu \nu }^{\sigma }=\frac{1}{2}{g}^{\sigma \lambda }({\partial }_{\mu }{g}_{\nu \lambda }+{\partial }_{\nu }{g}_{\lambda \mu }-{\partial }_{\lambda }{g}_{\mu \nu })$

These are calledChristoffel-symbols