In this work we discuss the deformed relativistic wave equations, namely the Klein-Gordon and Dirac equations in a Doubly Special Relativity scenario. We employ what we call a geometric approach, based on the geometry of a curved momentum space, which should be seen as complementary to the more spread algebraic one. In this frame we are able to rederive well-known algebraic expressions, as well as to treat yet unresolved issues, to wit, the explicit relation between both equations, the discrete symmetries for Dirac particles, the fate of covariance, and the formal definition of a Hilbert space for the Klein-Gordon case.

In this manuscript, we will discuss the notion of curved momentum space, as it arises in the discussion of noncommutative or doubly special relativity theories. We will illustrate it with two simple examples, the Casimir effect in anti-Snyder space and the introduction of fermions in doubly special relativity. We will point out the existence of intriguing results, which suggest nontrivial connections with spectral geometry and Hopf algebras.