Editing Covariant derivative (section)

==History==
Historically, at the turn of the 20th century, the covariant derivative was introduced by [[Gregorio Ricci-Curbastro]] and [[Tullio Levi-Civita]] in the theory of [[Riemannian geometry|Riemannian]] and [[pseudo-Riemannian manifold|pseudo-Riemannian geometry]].<ref>{{cite journal |last2=Levi-Civita |first2=T. |last1=Ricci |first1=G. |title=Méthodes de calcul différential absolu et leurs applications |url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PID=GDZPPN002258102 |journal=Mathematische Annalen |volume=54 |year=1901 |issue=1–2 |pages=125–201 |doi= 10.1007/bf01454201|s2cid=120009332 }}</ref> Ricci and Levi-Civita (following ideas of [[Elwin Bruno Christoffel]]) observed that the [[Christoffel symbols]] used to define the [[Riemann tensor|curvature]] could also provide a notion of [[derivative|differentiation]] which generalized the classical [[directional derivative]] of [[vector fields]] on a manifold.<ref>{{cite book |last=Riemann |first=G. F. B. |chapter=Über die Hypothesen, welche der Geometrie zu Grunde liegen |title=Gesammelte Mathematische Werke |year=1866 }}; reprint, ed. Weber, H. (1953), New York: Dover.</ref><ref>{{cite journal |last=Christoffel |first=E. B. |title=Über die Transformation der homogenen Differentialausdrücke zweiten Grades |journal=[[Crelle's Journal|Journal für die reine und angewandte Mathematik]] |volume=70 |year=1869 |pages=46–70 |url=https://eudml.org/doc/148073 }}</ref> This new derivative – the [[Levi-Civita connection]] – was ''[[Covariance and contravariance of vectors|covariant]]'' in the sense that it satisfied Riemann's requirement that objects in geometry should be independent of their description in a particular coordinate system.

It was soon noted by other mathematicians, prominent among these being [[Hermann Weyl]], [[Jan Arnoldus Schouten]], and [[Élie Cartan]],<ref>cf. with {{cite journal |last=Cartan |first=É |url=http://www.numdam.org/item?id=ASENS_1923_3_40__325_0 |title=Sur les variétés à connexion affine et la theorie de la relativité généralisée |journal= Annales Scientifiques de l'École Normale Supérieure|volume=40 |year=1923 |pages=325–412 |doi=10.24033/asens.751 |doi-access=free }}</ref> that a covariant derivative could be defined abstractly without the presence of a [[metric tensor|metric]]. The crucial feature was not a particular dependence on the metric, but that the Christoffel symbols satisfied a certain precise second-order transformation law. This transformation law could serve as a starting point for defining the derivative in a covariant manner. Thus the theory of covariant differentiation forked off from the strictly Riemannian context to include a wider range of possible geometries.

In the 1940s, practitioners of [[differential geometry]] began introducing other notions of covariant differentiation in general [[vector bundle]]s which were, in contrast to the classical bundles of interest to geometers, not part of the [[tensor analysis]] of the manifold. By and large, these generalized covariant derivatives had to be specified ''ad hoc'' by some version of the connection concept. In 1950, [[Jean-Louis Koszul]] unified these new ideas of covariant differentiation in a vector bundle by means of what is known today as a [[Koszul connection]] or a connection on a vector bundle.<ref>{{cite journal |last=Koszul |first=J. L. |title=Homologie et cohomologie des algebres de Lie |journal=Bulletin de la Société Mathématique de France |volume=78 |year=1950 |pages=65–127 |doi=10.24033/bsmf.1410 |doi-access=free }}</ref> Using ideas from [[Lie algebra cohomology]], Koszul successfully converted many of the analytic features of covariant differentiation into algebraic ones. In particular, Koszul connections eliminated the need for awkward manipulations of [[Christoffel symbols]] (and other analogous non-[[tensor]]ial objects) in differential geometry. Thus they quickly supplanted the classical notion of covariant derivative in many post-1950 treatments of the subject.