I have found a theorem in Frenkel's book attributed to Riemann which says:
In a Riemannian manifold, one can introduce local coordinates y such that the metric assumes the euclidean or flat form
\mathrm{d}s^2=(\mathrm{d}y^1)^2 + ... + (\mathrm{d}y^n)^2
if and only if curvature vanishes...
Thanks for help :) Let me get this straight - the general affine connection coefficients are defined by
\nabla_{\mathbf{e}_j}\mathbf{e}_k=\mathbf{e}_i \omega ^{i}_{jk}
where \mathbf{e}_i are the vectors of any basis. Orthonormality of the basis does not imply here that connection coefficients...
Hey!
If a (pseudo) Riemannian manifold has an orthonormal basis, does it mean that Riemann curvature tensor vanishes? Orthonormal basis means that the metric tensor is of the form
(g_{\alpha\beta}) = \text{diag}(-1,+1,+1,+1)
what causes Christoffel symbols to vanish and puts Riemann...