It is often said that mass curves space-time, but interestingly there is more than one definition of curvature and only one is what is meant by that statement. Curvature in terms of what is usually called Riemanian space-time curvature is one definition. Another definition of curvature as what is usually called Gaussian curvature . I'll demonstrate the differences.
First lets consider the definition for curvature given by Gaussian curvature.
Here's a classic thought experiment. Imagine a two dimensional creature living on the surface of a sphere. But to its surprise measures that C is less than 2 p r. After careful thought it realized that its two dimensional world must be a positively curved space imbedded in a third dimension. From here we can make a more general definition of curvature in terms of excess radius .