Number theory has been used to prove many interesting results on $SO(3)$ and related Lie groups, which in turn has attractive applications for the underlying symmetric spaces. A prime example is Drinfeld's solution of the Ruziewicz problem on invariant means of the sphere, or the related recent work of Bourgain-Gamburd on the spectral gap for finitely generated subgroups of $SU(2)$. Related is the Banach-Tarski paradox on doubling the ball, or the recent result of Kiss-Laczkovich that a ball can be decomposed into 22 (or more) congruent pieces.
I also regard Gödel's incompleteness theorem as an application of number theory. Some variants of it, like Matiyasevich's theorem on diophantine equations is highly number theoretic both in its statement and proof.
