There is a geometric picture of the electromagnetic field in terms of "warps and curves," as you put it. This picture is in precise analogy to that general relativity, although it involves different mathematical objects with different physical interpretations.
In general relativity, the "warps and curves" in spacetime are described mathematically by what is called the curvature tensor of spacetime. With the curvature tensor, we can determine the gravitational force on particles. Einstein's equations tell us that the curvature tensor is related to the distribution of matter and energy in spacetime.
In electromagnetism, we can relate electric and magnetic forces to a curvature tensor in a similar way. However, the curvature tensor in the case of electromagnetism is a fundamentally different object both mathematically and physically. It does not describe the curvature of spacetime. Instead, it describes the curvature of a kind of "internal" space that is attached to the points of spacetime. (Mathematically, this "internal" space is a called a fiber bundle. Physically, the study of such spaces is called gauge theory.)
Just like Einstein's equations relate the curvature of spacetime to the presence of energy and matter, Maxwell's equations relate the electromagnetic curvature to the presence of electromagnetic charges and currents.