synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
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.
Discussion of normal coordinates in supergeometry (super Cartan geometry):
with application to sigma-models
Joseph Atick, Avinash Dhar, §3 in: -Expansion and light-cone gauge fixing in curved superspace -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:
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.
Last revised on June 27, 2024 at 08:31:27. See the history of this page for a list of all contributions to it.