List of Formulas in Riemannian Geometry - in An Inertial Frame

In An Inertial Frame

An orthonormal inertial frame is a coordinate chart such that, at the origin, one has the relations and (but these may not hold at other points in the frame). In such a frame, the expression for several operators is simpler. Note that the formulae given below are valid at the origin of the frame only.

R_{ik\ell m}=\frac{1}{2}\left( \frac{\partial^2g_{im}}{\partial x^k \partial x^\ell} + \frac{\partial^2g_{k\ell}}{\partial x^i \partial x^m} - \frac{\partial^2g_{i\ell}}{\partial x^k \partial x^m} - \frac{\partial^2g_{km}}{\partial x^i \partial x^\ell} \right)

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