Manifolds, Normal Frames and Riemannian Coordinates

9. Conclusion

This chapter, as we saw, has an introductory character. It does not contain new original material except the implicit description of the frames normal at a single point of a Riemannian manifold (Section 6) and partially the investigation of normal frames/coordinates in Section 7.

After the presentation of the minimum knowledge from the differential geometry, required for our work, we started with the initial ideas concerning normal frames and coordinates. The basic results here are: only torsionless linear connections (may) admit normal coordinates; if the torsion is non-zero, normal frames (may) exit, but normal coordinates do not. If normal frames exist, they are parallel and are connected with linear transformations whose matrices are constant under the action of their basic vector fields.

The Riemannian and geodesic coordinates, which are normal at their origins, were pointed out as first examples of normal coordinates.