Curvatures of spacetime

Curvatures of spacetime is a set of 16 various curvatures that may be specified as components of a curvature tensor that describes the curvature of spacetime (the same thing as curvature of 4-space).

Curvature of spacetime is something that describes a fact that some 4-dimensional space is curved which in our world is a myth when applied to our universe since its spacetime is flat (or Minkowski, which is another word for the same thing). In particular it may describe also the curvature of the common 3-dimenssional space (given by x,y,z vectors) being a part of this 4-dimensional spacetime (in this case given by t,x,y,z 4-vectors), which is a chance for messing things in cosmology when one says "curvature of space" but means the "curvature of spacetime" (that is then called simply curvature of 4-space). It happens a lot in cosmology when astronomers customarily mix curvature of space (the obvious feature of 3-space) with curvature of spacetime that has to be flat (otherwise known as Minkowski), though having its non zero curvatures of spacetime adding to the flatness of whole spacetime.

The 4-space has to be flat since parallel transport of 4-vectors along its geodesic lines can't change the angles of those vectors for the reason of conservation of energy (the 4-vectors have to come back after the transport being directd in the same direction as before the transport since some of their components as energy or momentum in various directions must be conserved (by Loether theorem).

That's why it is often important to be aware of various subtleties of names used in physics not to violate suddenly the conservation of energy and/or momentum, since so far the nature does not know how to make energy or momentum from nothing.