  Several arguments suggest that the Planck scale could be the characteristic
scale of curvature of momentum space. As other recent studies we assume that
the metric of momentum space determines the condition of on-shellness while the
momentum-space affine connection governs the form of the law of composition of
momenta. We show that the possible choices of laws of composition of momenta
are more numerous than the possible choices of affine connection on a momentum
space. This motivates us to propose a new prescription for associating an
affine connection to momentum composition, which we compare to the one most
used in the recent literature. We find that the two prescriptions lead to the
same picture of the so-called $\kappa$-momentum space, with de Sitter metric
and $\kappa$-Poincar\'e connection. We also examine in greater detail than ever
before the DSR-relativistic properties of $\kappa$-momentum space, particularly
in relation to its noncommutative law of composition of momenta. We then show
that in the case of "proper de Sitter momentum space", with the de Sitter
metric and its Levi-Civita connection, the two prescriptions are inequivalent.
Our novel prescription leads to a picture of proper de Sitter momentum space
which is DSR-relativistic and is characterized by a commutative law of
composition of momenta, a possibility for which no explicit
curved-momentum-space picture had been previously found. We argue that our
construction provides a natural test case for the study of momentum spaces with
commutative, and yet deformed, laws of composition of momenta. Moreover, it can
serve as laboratory for the exploration of the properties of DSR-relativistic
theories which are not connected to group-manifold momentum spaces and Hopf
algebras.
