Electromagnetism as curvature similar to gravity? My knowledge of Physics is very surface-level, so sorry if this question doesn't make sense. Before Einstein's theory of relativity, people would think of the gravitational field as a force exerted by massive objects. Now we have a new interpretation where mass simply curves space-time and it is the movement of objects through the curved spacetime that causes the illusion of a force.
Now, the Electromagnetic force looks a lot like the gravitational force, with both following an inverse square law. Is it possible then to interpret the electromagnetic force as the consequence of movement through a curved surface as well? Is there any attempt at this already?
 A: Yes indeed, as pointed out by many responders in their comments here. Kaluza found that if Einstein's equations for general relativity in three dimensions of space and one of time were rewritten to include an extra spatial dimension, he could obtain from that Maxwell's equations for electrodynamics, which Einstein considered a significant breakthrough. Later, Klein proposed the idea that the extra dimension had the shape of a circle and was compactified, rendering it invisible to us.
It was also demonstrated that by rewriting Maxwell's equations in five dimensions instead of four, one could mathematically extract the equations of general relativity from the result.
However, these formulations all contained features that did not square with the real world and which could not be fixed in the math. So as appealing as the idea may have been initially, Kaluza-Klein theory wound up being a dead end as far as unifying gravity with electromagnetism- but the basic idea of "extra" compactified dimensions with complex topologies lives on in string theory.
