# Did the formation of Differential Geometry come before Differential Topology/ Topology in general?

I’m pretty interested in the history of mathematics, and it has always been my belief that the great pioneers of Differential Geometry were Gauss and Riemann, and the father of topology was mostly accredited to Poincare. However, now I am taking a differential topology/geometry sequence and we basically learn two semesters of Differential Topology as a prerequisite for Differential Geometry, and this confuses me since Riemann and Gauss were both dead before Poincare could’ve done anything with topology. So was Differential Geometry created before Differential Topology? If so how were Gauss and Riemann able to do it before what seems to be the necessary definitions of topological spaces, and smooth manifolds?

Also if anyone has any sources on a history of modern mathematics (like 18-21st century) that they could recommend, that would be very cool since I really can’t seem to find any that really get into the weeds as much as I would like.

• The order of teaching is not the order of history. Euler did differential geometry of curves and surfaces before Gauss, and he was not "the first" either, and one sure does not need abstract theory of manifolds to handle curves and surfaces. Concrete examples only deal with instances of abstract ideas, which are "there" only in hindsight, and from those they crystallize and differentiate into a separate subject. But it often saves time and effort to teach in reverse. Sep 16, 2021 at 3:14
• @Conifold I would like to know how they were able to it without topology though Sep 16, 2021 at 3:17
• @ChristopherQuinnLaFondJr. Why do you think Euler and Gauss needed topology for this? They needed calculus which was already available. Sep 16, 2021 at 4:04
• Remember how they find curvatures of curves and surfaces in calculus textbooks without calling them manifolds? For historical surveys see e.g. Coolidge, A History of Geometrical Methods, book III, Dombrowski, 150 years after Gauss, Dieudonné, History of Algebraic and Differential Topology. Sep 16, 2021 at 4:07
• Sep 16, 2021 at 4:26