Curvatures of spacetime
Curvatures of spacetime (note the plural) is a set of 10 various curvatures that may be specified as components of a curvature tensor that describes the curvature of spacetime (the same thing as the curvature of 4-space).
Curvature of spacetime (singular as opposed to plural "curvatures") is something that describes a fact that some 4-dimensional space is curved. It can't happen though in a stationary universe since the spacetime of stationary universe has to be flat (or Minkowski, which is another word for the same thing). In particular the curvatures of spacetime may describe the curvature of 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 astronomy when one says the "curvature of space" but means the "curvature of spacetime" (that is then called simply the curvature of 4-space) that has to be flat in stationary universe though having its non zero curvatures of spacetime adding to the flatness of whole spacetime.
The 4-space has to be flat in a stationary universe since parallel transport of 4-vectors along geodesic lines can't change the angles of those vectors for the reason of conservation of energy and momentum: the 4-vectors have to come back after the parallel transport being directd in the same direction and with the same magnitudes as before the transport by Noether's theorem. To keep energy and momentum conserved there can't be any permanent changes in components of those vectors between the now and the past or the future.
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 by requiring "gravity to happen in the curved spacetime", while it is not required by any physical or geometrical law as Einsteinian gravitation might be happening just between particles interacting on a quantum level as described in the gravitation demystified.