The hyperfine spectra of Mn51,53-64 were measured in two experimental runs using collinear laser spectroscopy at ISOLDE, CERN. Laser spectroscopy was performed on the atomic 3d54s2S5/26→3d54s4pP3/26 and ionic 3d54s5S2→3d54p5P3 transitions, yielding two sets of isotope shifts. The mass and field shift factors for both transitions have been calculated in the multiconfiguration Dirac-Fock framework and were combined with a King plot analysis in order to obtain a consistent set of mean-square charge radii which, together with earlier work on neutron-deficient Mn, allow the study of nuclear structure changes from N=25 across N=28 up to N=39. A clear development of deformation is observed towards N=40, confirming the conclusions of the nuclear moments studies. From a Monte Carlo shell-model study of the shape in the Mn isotopic chain, it is suggested that the observed development of deformation is not only due to an increase in static prolate deformation but also due to shape fluctuations and triaxiality. The changes in mean-square charge radii are well reproduced using the Duflo-Zuker formula except in the case of large deformation.