Now I see the term _metric spacetime_ in the book __Teleparallel Gravity__ - 'Curvature and torsion are tensorial properties of Lorentz connections. This becomes evident if we note that many different connections can be defined on the very same _metric spacetime_. Of course, when restricted to the specific case of General Relativity, where only the zero-torsion spin connection is present, universality
of gravitation allows its curvature to be interpreted—together with the metric—as
part of the spacetime definition, and one can then talk about “spacetime curvature”.'