Can I get a PhD in artificial intelligence with a pure mathematics background?

You have some of the elements of an artificial intelligence major. Your background in logic is pertinent. And your background in mathematical analysis and Riemannian geometry is at least interesting.

What's troubling about what I've seen of your background is your lack of computer science courses. Maybe that's not an issue, if you have the equivalent of a minor in computer science that wasn't featured in your question. Certainly, that would be an important "tool." And if you don't have it, you might consider going back to school for a five or six course minor or certificate program in CS.

The most interesting part of your question was, "I find the Artificial Intelligence field really fascinating." Why is that? And how does that connect with the rest of your background? Or are you starting from scratch, which would be difficult, though not impossible. In any event, these issues should be addressed in your statement of purpose.

Here's my experience in unfamiliar territory: In high school, I had four years of French, and none of Spanish (unless you count my one year of self-taught Spanish). In college, I bluffed my way into a second year Spanish class, and completed it successfully. Here's an example where my interest and natural aptitude compensated for my lack of formal training. Perhaps it will be the same for you and artificial intelligence.

The short answer: definitely yes, if you pick the right topic.

The longer answer: assuming you can catch up with some core programming based on Matlab or numpy/scipy, with your background you should have very good chances in theoretically oriented Machine Learning groups. Riemannian geometry will put you in a very good position for instance to pursue Information Geometry, which is a branch of information theory/probability theory of high relevance to learning theory which makes heavy use of differential geometry.

I know of a student with with a pure math background who continued to do a PhD in AI, with great success and strong publications. He was a good programmer, though, without actually having studied CS formally.

I just want to supplement other answers (I have a PhD in AI). If your background is mathematics, I do not agree with others that you need to code per-se. Aside from machine learning pointed above, there are some other sides you can look at that do not require - much - programming (if any).

Logic - Symbolic AI is broadly focused around temporal logics, including those specifically formalised for game-theory, and non-monotonic logics. You can go in many directions with logic in AI: philosophical (i.e., thinking of how to capture certain types of common-sense reasoning, normative reasoning, etc.) or more solution-oriented (i.e., we need a logic to represent a specific problem X and then we will prove some general results about that logic). In the case that you work on logics to solve specific problems, you may need some Computer Science understanding (in particular, proving complexity for this kind of research is required for publishing in many top venues). In terms of the more philosophical aspects, you don't really need so much of a Comp. Sci. background, non-monotonic logics, for example, can be generalised in a graph-theoretical framework.

Game theory - similar, to the problem-oriented logic research I discussed previously, many papers in AI capture some new game-theoretical concepts and aim to prove something. In this area are related topics, such as negotiation (and to some extent, argumentation, although that is more related with non-monotonic logic). For some game-theoretical papers that relax assumptions of heterogeneity in the participants involved in the game (for example), simulation is used to find support for properties that cannot feasibly be proven (often, properties with assumptions of homogonous participants are proven and simulation supplements these results with experiments on trickier cases).

In short, I see no problem with you moving to AI, we have plenty of mathematicians (and philosophers and computer scientists) in the field. In terms of logic or game-theory, you may want to pick the topic carefully if you want to avoid spending time learning 'theoretical' Comp. Sci and/or programming.