A. Einstein, The Foundations of the General Theory of Relativity,from the Principle of Relativity,Dover(1952);section D Material Phenomena §20 Maxwell's Equations for Free Space -- back to the source
R. C. Tolman, Relativity, Thermodynamics and Cosmology,Dover(1987);general heading VIII Relativistic Electrodynamics,parts I&II,chapters §101-§116 -- oldie but goodie
C. Misner,K. Thorne and J. Wheeler, Gravitation,Freeman(1973);part IV Einstein's Geometric Theory of Gravity,chapter 22 Thermodynamics, Hydrodynamics, Geometric Optics and Kinetic Theory,§22.4 Electrodynamics in Curved Space -- a newer classic
The last two items might require reading earlier parts in the books about electromagnetic theory in non-curved minkowskian spacetime in order to get the notation and general drift of the authors down.
Mind you, these references won't attempt to explain electromagnetism purely in terms of curved spacetime; they just make room for an electromagnetic field in General Relativity.
Kaluza-Klein is not accepted as a genuine theory of EM in curved spacetime. However the basic idea of K-K, compact extra dimensions, has been adopted by string theory.
The problem with K-K as a basic theory of EM is that it is classical, and physicists regard the true basic theory of EM to be quantum. They have a hierarchy of "effective theories" with classical EM at the bottom, then non-relativistic QM, then Dirac's relativistic electron theory, then Quantum Electrodynamics (QED, the theory that Feynman, Schwinger and Tomonaga got the Nobel for), then electroweak theory (part of the standard model, Weinberg, Feinberg, and Salam got the Nobel for that one).
Even Einstein, who had just introduced the quantized EM field (the photon) seems to have realized that building classical EM into GR was a lost cause.
Nevertheless EM is still an effective theory. For low energy or slow moving potentials it is an excellent calculating tool, and that is why there are all those textbooks on how to set it up in curved spacetime.
The historical topic of unification of gravitation with electromagnetism in the Einstein direction takes one down a trail that hasn't survived the explosion of theory for fundamental physics since 1960. But for a while, it was almost Einstein and his associates alone who credibly pushed the idea of unification with gravitation. That was because of Einstein's enormous reputation. But time ran out on this line.
P. Bergmann, Introduction to the Theory of Relativity,Dover(1976); Part III Unified Field Theories, chapters XVI-XVIII -- this describes the Weyl theory and Kaluza 5D theory that first sent Einstein down the path of unification. But Einstein was not happy with the extra space dimension and tried for a kind of projection into standard spacetime.
A. Einstein,The Meaning of Relativity,5th edition,Princeton(1972); appendix II Relativistic Theory of the Non-symmetric Field -- Einstein's greatest effort at unification was a non-symmetric fundamental tensor approach, where the gij quantities are no longer assumed to be symmetric and the spacetime structure is post-riemannian affine-connected. He even declared a victory of sorts by 1949, but it was not clearly so to others.
A. Pais,'Subtle is the Lord...',Oxford(1982); chapter 17 Unified Field Theory -- this is a good survey of the various approaches Einstein tried for unification of electromagnetism with gravitation
E. Schrödinger,Space-Time Structure,Cambridge(2002); chapter XII Generalizations of Einstein's Theory -- Schrödinger and Einstein were united in opposition to a purely statistical and probabilistic foundation to the quantum. Schrödinger had tried originally for a 4D equation, but decided it was wrong and published his quasi-classical energy equation instead, thereafter known as the Schrödinger wave equation. Einstein could not stand 4D versions of quantum theory at all. In this monograph, ES reports some of the work of Einstein's unified field collaborators. It is such a nice book, it belongs with the clear and utterly convincing presentations of the Tolman book I listed in my earlier post.
In the general relativity renaissance of the 1960s, just about everyone agreed that quantum fields were the right starting point for any unification efforts, thereby passing up the classical electromagnetism question altogether.
I found an article referring to the electromagnetic theory in curved spacetime in the databese of preprint server. In the article, non-guage-invariant theory of electromagnetism is deduced from Kaluza-Klein theory. Is it possible that broken symmetry of guage invariance by spacetime curvature makes electromagnetc source appear like broken symmetry of unitary invariance (I mean unitary inequivalent vacua appear) leads to Unruh radiation or Hawking radiation in quantum field theory in curved spacetime, if my understanding is correct?