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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.

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Yes, differential geometry is older. Though he had predecessors, Gauss can be considered the founding father of differential geometry with his book General investigation of curved surfaces, 1827. This was long before the subject of topology was born as a subject (though some isolated results in both differential geometry and topology are much older). Systematic development of topology is usually credited to a series of papers of Poincare published in 1893-1904. This is about systematic development of the two disciplines.

But when you look at the earliest isolated results, the situation is similar: Differential Calculus was applied to geometry (of curves) since its very beginning of calculus, while the first recorded theorem of topology is probably Euler's Konigsberg bridges problem.

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