Einstein's General Theory of Relativity construes space as three dimensions: as three axes of four-dimensional space-time. The notion that gravity is the curvature of space-time originates with Einstein himself. However the curvature that Einstein described is actually the curvature of a trajectory through space, as in the case of light passing a massive body.
What is actually curved is the geodesic: the shortest path between two points. The reason why the trajectory is curved is that, according to General Relativity, gravity is the contraction of space intervals in the direction of the centre of mass. This means that space intervals along these vectors are contracted relative to space intervals along perpendicular spatial axes. Consequently, the shortest trajectory of light in three-dimensional space is bent in the direction of the centre of mass.
The misunderstanding of space — rather than geodesics — as curved leads to misunderstandings like the following, where gravity is misrepresented as a deformation of otherwise flat space, instead of as the increasing contraction of space intervals along all vectors in the direction of the centre of mass:
