There is also another use of the term curvature. But I suspect that it is mostly avoided nowadays and appears only in Einstein's original work. When Einstein used the term curvature he sometimes (all the time?) meant for it to mean acceleration. This is how he used it in his book Relativity: The Special And General Theory. A related term to accelerated motion is curvilinear motion. What is a straight line in one frame is curvilinear in another.