I wish to calculate connection, curvature, Ricci curvature etc. for a
Riemannian metric given as follows: there is an orthogonal frame of vector
fields with stipulated Lie bracket relations between them. The frame is
orthogonal but not orthonormal, and the lengths of its vector fields are functions
of a single function on the manifold. Given these metric values on the frame and the
Lie bracket relations, the covariant derivatives are in principle computable from the
Koszul formula, hence connection and curvature are all determined.
When I try to define the metric using a dual coframe in ATLAS's Metric
routine, it allows me to define it but claims there is not actual curvature.
From the help it seems the coframes used in this routine are always given
as differentials of coordinates. Is there a way to get the metric via the data
given above without putting in by hand all the different Koszul formulas etc.?