No, a magnetic field is not generated by the curvature of space. The curvature of space is a concept in general relativity that describes how the presence of mass and energy can curve the fabric of spacetime, affecting the motion of objects and the propagation of light.
On the other hand, a magnetic field arises due to the movement of electric charges. When electric charges, such as electrons, are in motion, they generate a magnetic field around them. This is described by electromagnetic theory, specifically by Ampere's law and the Biot-Savart law.
While it is true that the presence of mass and energy can affect the behavior of electromagnetic fields, as described by general relativity, the generation of magnetic fields is fundamentally a result of moving electric charges and not directly related to the curvature of space.