By "geometric" I'm going to assume that you mean "having to do with the geometry of the usual 3+1 dimensions", that is, geometric in the sense that electricity and magnetism is geometric.
This is a question that was researched deeply in the 1950s especially by Coleman and Mandula after which the "Coleman-Mandula" theorem is named. As wikipedia puts it, "It states that "space-time and internal symmetries cannot be combined in any but a trivial way". Thus the internal symmetries are not related to "geometry" in the sense of our usual world.
The Coleman-Mandula theorem depends on a lot of complicated mathematics. One can imagine a bunch of ways that one could get around it. If you google Coleman-Mandula on arXiv.org, you can find papers on this subject, i.e. extensions and consequences of the Coleman-Mandula theorem, as well as papers proposing how one might get around it: