nLab Riemann normal coordinates

Idea

Around every point of a Riemannian manifold there is a coordinate system such that the Levi-Civita connection of the metric pulled back to these coordinates vanishes at that point. (Notice that the Riemann curvature will not in general vanish even at that point).

In the context of general relativity this reflects aspects of the equivalence principle (physics).

In the sense of integrability of G-structures, Riemann normal coordinates exhibit the first-order integrability of orthogonal structure, see at integrability of G-structures – Examples – Orthogonal structure.

Related concepts

• general covariance

• Walker coordinates

• Wess-Zumino gauge

References

General

• Wikipedia, Normal coordinates

In supergeometry

Discussion of normal coordinates in supergeometry (super Cartan geometry):

• Ian N. McArthur, Superspace normal coordinates, Class. Quantum Grav. 1 (1984) 233 [doi:10.1088/0264-9381/1/3/003, inspire:13800, pdf]

with application to sigma-models

• Joseph Atick, Avinash Dhar, §3 in: $\theta$-Expansion and light-cone gauge fixing in curved superspace $\sigma$-models, Nuclear Physics B 284 (1987) 131-145 [doi:10.1016/0550-3213(87)90029-0]

• Sylvester James Gates Jr., Marcus Grisaru, Marcia E. Knutt-Wehlau, Warren Siegel, Component actions from curved superspace: Normal coordinates and ectoplasm, Physics Letters B 421 1–4, 5 (1998) 203-210 [doi:10.1016/S0370-2693(97)01557-8]

and to supergravity:

• Marcus Grisaru, Marcia E. Knutt-Wehlau, Warren Siegel, A Superspace normal coordinate derivation of the density formula, Nucl. Phys. B 523 (1998) 663-679 [doi:10.1016/S0550-3213(98)00151-5, arXiv:hep-th/9711120]

specifically to D=11 supergravity:

• Dimitrios Tsimpis, Curved 11D supergeometry, JHEP11 (2004) 087 [doi:10.1088/1126-6708/2004/11/087, arXiv:hep-th/0407244]

  where a version of fermionic Riemann normal coordinates are thought of as the “most natural generalization” of Wess-Zumino gauge.