  Context. While rotation has a major impact on stellar structure and
evolution, its effects are not well understood. Thanks to high- quality and
long timebase photometric observations obtained with recent space missions, we
are now able to study stellar rotation more precisely. Aims. We aim to
constrain radial differential rotation profiles in gamma Doradus (gamma Dor)
stars, and to develop new theoretical seismic diagnosis for such stars with
rapid and potentially non-uniform rotation. Methods. We derive a new asymptotic
description which accounts for the impact of weak differential near-core
rotation on gravity- mode period spacings. The theoretical predictions are
illustrated from pulsation computations with the code GYRE and compared with
observations of gamma Dor stars. When possible, we also derive the surface
rotation rates in these stars by detecting and analysing signatures of
rotational modulation, and compute the core-to-surface rotation ratios.
Results. Stellar rotation has to be strongly differential before its effects on
period spacing patterns can be detected, unless multiple period spacing
patterns can be compared. Six stars in our sample exhibit a single unexplained
period spacing pattern of retrograde modes. We hypothesise that these are Yanai
modes. Finally, we find signatures of rotational spot modulation in the
photometric data of eight targets. Conclusions. If only one period spacing
pattern is detected and analysed for a star, it is difficult to detect
differential rotation. A rigidly rotating model will often provide the best
solution. Differential rotation can only be detected when multiple period
spacing patterns have been found for a single star or its surface rotation rate
is known as well. This is the case for eight stars in our sample, revealing
surface-to-core rotation ratios between 0.95 and 1.05.
