Bernhard Riemann's famous 1854 Habilitation lecture "On the hypotheses…" owed more to the influence of the anti-Kantian philosopher Johann Friedrich Herbart than to Kant himself. Here one could mention an illuminating article by Nowak:
G. Nowak, in The history of modern mathematics, Vol. I (Poughkeepsie, NY, 1989), 17–46, Academic Press, Boston, MA, 1989.
Nowak argues that
Herbart's constructive approach to space, already cited, mirrored the content of Riemann's reference to Gauss in that both discussed construction of spaces rather than construction in space.
Riemann followed Herbart in rejecting Kant's view of space as an a priori category of thought, instead seeing space as a concept which possessed properties and was capable of change and variation. Riemann copied some passages from Herbart on this subject, and the Fragmente philosophischen Inhalts included in his published works contain a passage in which Riemann cites Herbart as demonstrating the falsity of Kant's view.
Riemann took from Herbart the view that the construction of spatial objects was possible in intuition and independent of our perceptions in physical space. Riemann extended this idea to allow for the possibility that these spaces would not obey the axioms of Euclidean geometry. We know from Riemann's notes on Herbart that he read Herbart's Psychologie als Wissenschaft.
It is possible that, had Kant's ideas about infinite space not been challenged by Herbart, the field we know today as Riemannian geometry may have developed much later.
This is meant to complement Thomas Klimpel's answer on Riemann.
For those with access to mathscinet, more information can be found here.
Note. Peter Heinig pointed out that the historian David Rowe similarly holds that Riemann was more of a Herbartian than Kantian; see comments below and also Rowe's review.
