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.
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