From the point of view of modern physics and the standard model we describe the three forces of Electromagnetism, Weak, and Strong nuclear forces using the exchange of force field quanta. This is where the idea of a "force-carrier" particle comes in. The Photon, Weak Boson, and Gluon. In a quantum theory of gravity thus would be the graviton.

However, your idea about using geometry to describe other forces is not off the mark. This was the path pursued by theorists before the formalization of quantum theory in the 1920s. Kaluza-Klein theory adds extra dimensions to space-time and manages to derive a description of EM fields using the extra pieces of the metric tensor. There are other extensions of gravity that use the torsion field and with this you get more degrees of freedom to play with (even without extra dimensions). Einstein himself started research in both of these areas, add torsion to his final edition of relativity before his death.

Modern particle physics is very successful in describing everything but gravity. The paradigm there is that the force fields arise from local symmetry of the matter fields. This is called gauge theory and uses group theory, symmetries and Lie algebra to describe the pattern we see in particle data. GR uses differential geometry to describe the nature of gravity. Now there is symmetry in gravity, group theory does come into play, and there is an underlying differential geometry for gauge group manifolds. However, they mean different things in each paradigm and are somewhat at odds with each other. Most QF theorists and particle physicists I learned form in graduate school vehemently opposed unification that didn't respect the standard model, thinking GR was an "ugly" theory. I went the other way. I think there is a lot of uncharted territory in the extensions of differential geometry that have not been ruled out. Then of course there is string theory.

From the point of view of modern physics and the standard model we describe the three forces of Electromagnetism, Weak, and Strong nuclear forces using the exchange of force field quanta. This is where the idea of a "force-carrier" particle comes in. The Photon, Weak Boson, and Gluon. In a quantum theory of gravity thus would be the graviton.

However, your idea about using geometry to describe other forces is not off the mark. This was the path pursued by theorists before the formalization of quantum theory in the 1920s. Kaluza-Klein theory adds extra dimensions to space-time and manages to derive a description of EM fields using the extra pieces of the metric tensor. There are other extensions of gravity that use the torsion field and with this you get more degrees of freedom to play with (even without extra dimensions). Einstein himself started research in both of these areas, add torsion to his final edition of relativity before his death.

Modern particle physics is very successful in describing everything but gravity. The paradigm there is that the force fields arise from local symmetry of the matter fields. This is called gauge theory and uses group theory, symmetries and Lie algebra to describe the pattern we see in particle data. GR uses differential geometry to describe the nature of gravity. Now there is symmetry in gravity, group theory does come into play, and there is an underlying differential geometry for gauge group manifolds. However, they mean different things in each paradigm and are somewhat at odds with each other. Most QF theorists and particle physicists I learned from in graduate school vehemently opposed unification that didn't respect the standard model, thinking GR was an "ugly" theory. I went the other way. I think there is a lot of uncharted territory in the extensions of differential geometry that have not been ruled out. Then of course there is string theory.