- History of manifolds and varieties
The study of

manifold s combines many important areas ofmathematics : it generalizes concepts such ascurve s andsurfaces as well as ideas fromlinear algebra andtopology . Certain special classes of manifolds also have additional algebraic structure; they may behave like groups, for instance. In that case, they are calledLie Group s.**Background**Ancestral to the modern concept of a manifold were several important results of eighteenth and

nineteenth century mathematics. The oldest of these wasNon-Euclidean geometry , which considers spaces whereEuclid 'sparallel postulate fails. Saccheri first studied this geometry in1733 . Lobachevsky, Bolyai, and Riemann developed the subject further 100 years later. Their research uncovered two types of spaces whose geometric structures differ from that of classicalEuclidean space ; these are calledhyperbolic geometry andelliptic geometry . In the modern theory of manifolds, these notions correspond to manifolds with constant, negative and positivecurvature , respectively.Carl Friedrich Gauss may have been the first to consider abstract spaces as mathematical objects in their own right. Histheorema egregium gives a method for computing thecurvature of asurface without considering theambient space in which the surface lies. In modern terms, the theorem proved that the curvature of the surface is an intrinsic property. Manifold theory has come to focus exclusively on these intrinsic properties (or invariants), while largely ignoring the extrinsic properties of the ambient space.Another, more topological example of an intrinsic property of a manifold is the

Euler characteristic . For a non-intersecting graph in the Euclidean plane, with "V" vertices (or corners), "E" edges and "F" faces (counting the exterior) Euler showed that "V"-"E"+"F"= 2. Thus 2 is called the Euler characteristic of the plane. By contrast, in 1813 Antoine-Jean Lhuilier showed that the Euler characteristic of thetorus is 0, since thecomplete graph on seven points can be embedded into the torus. The Euler characteristic of other surfaces is a usefultopological invariant , which has been extended to higher dimensions usingBetti number s. In the mid nineteenth century, theGauss–Bonnet theorem linked the Euler characteristic to the Gaussian curvature.Lagrangian mechanics andHamiltonian mechanics , when considered geometrically, are naturally manifold theories. All these use the notion of several characteristic axes ordimension s (known asgeneralized coordinates in the latter two cases), but these dimensions do not lie along the physical dimensions of width, height, and breadth.In the early

nineteenth century the theory ofelliptic function s succeeded in giving a basis for the theory ofelliptic integral s, and this left open an obvious avenue of research. The standard forms for elliptic integrals involved thesquare root s of cubic andquartic polynomial s. When those were replaced by polynomials of higher degree, say quintics, what would happen?In the work of

Niels Abel and Carl Jacobi, the answer was formulated: this would involve functions oftwo complex variables , having four independent "periods" (i.e. period vectors). This gave the first glimpse of an**abelian variety**of dimension 2 (an**abelian surface**): what would now be called the "Jacobian of ahyperelliptic curve of genus 2".**Riemann**Bernhard Riemann was the first to do extensive work generalizing the idea of a surface to higher dimensions. The name "manifold" comes from Riemann's original German term, "Mannigfaltigkeit", whichWilliam Kingdon Clifford translated as "manifoldness". In his Göttingen inaugural lecture, Riemann described the set of all possible values of a variable with certain constraints as a "Mannigfaltigkeit", because the variable can have "many" values. He distinguishes between "stetige Mannigfaltigkeit" and "diskrete" "Mannigfaltigkeit" ("continuous manifoldness" and "discontinuous manifoldness"), depending on whether the value changes continuously or not. As continuous examples, Riemann refers to not only colors and the locations of objects in space, but also the possible shapes of a spatial figure. Using induction, Riemann constructs an "n-fach ausgedehnte Mannigfaltigkeit" ("n times extended manifoldness" or "n-dimensional manifoldness") as a continuous stack of (n−1) dimensional manifoldnesses. Riemann's intuitive notion of a "Mannigfaltigkeit" evolved into what is today formalized as a manifold.Riemannian manifold s andRiemann surface s are named after Bernhard Riemann.In 1857, Riemann introduced the concept of

Riemann surface s as part of a study of the process ofanalytic continuation ; Riemann surfaces are now recognized as one-dimensional complex manifolds. He also furthered the study of abelian and other multi-variable complex functions.**Contemporaries of Riemann**Johann Benedict Listing , inventor of the word "topology ", wrote an 1847 paper "Vorstudien zur Topologie" in which he defined a "complex". He first defined theMöbius strip in 1861 (rediscovered four years later by Möbius), as an example of anon-orientable surface .After Abel, Jacobi, and Riemann, some of the most important contributors to the theory of abelian functions were Weierstrass, Frobenius, Poincaré and Picard. The subject was very popular at the time, already having a large literature. By the end of the 19th century, mathematicians had begun to use geometric methods in the study of abelian functions.

**Poincaré**Henri Poincaré 's 1895 paper Analysis Situs studied three-and-higher-dimensional manifolds, giving rigorous definitions of homology, homotopy (which had originally been defined in the context of late nineteenth-centuryknot theory , developed by Maxwell and others), and Betti numbers and raised a question, today known as thePoincaré conjecture , based his new concept of thefundamental group . As of 2006, a consensus among experts is that recent work byGrigori Perelman may have answered this question, after nearly a century of effort by many mathematicians.**Later developments**Hermann Weyl gave an intrinsic definition for differentiable manifolds in 1912. During the 1930sHassler Whitney and others clarified the foundational aspects of the subject, and thus intuitions dating back to the latter half of the 19th century became precise, and developed throughdifferential geometry andLie group theory.Eventually, in the

1920s , Lefschetz laid the basis for the study of abelian functions in terms of complex tori. He also appears to have been the first to use the name "abelian variety "; inRomance languages , "variety" was used to translate Riemann's term "Mannigfaltigkeit". It was Weil in the1940s who gave this subject its modern foundations in the language of algebraic geometry.**Sources*** Riemann, Bernhard, [

*http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Grund/ "Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse"*] .

**The 1851 doctoral thesis in which "manifold" ("Mannigfaltigkeit") first appears.

* Riemann, Bernhard, [*http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/ "On the Hypotheses which lie at the Bases of Geometry"*] .

**The famous Göttingen inaugural lecture (Habilitationsschrift) of 1854.

* [*http://www-groups.dcs.st-andrews.ac.uk/~history/HistTopics/Knots_and_physics.html Early history of knot theory at St-Andrews history of mathematics website*]

* [*http://www-groups.dcs.st-andrews.ac.uk/~history/HistTopics/Topology_in_mathematics.html Early history of topology at St. Andrews*]

* H. Lange and Ch. Birkenhake,**Complex Abelian Varieties**, 1992, ISBN 0-387-54747-9

** A comprehensive treatment of the theory of abelian varieties, with an overview of the history the subject.

*André Weil :**Courbes algébriques et variétés abéliennes**, 1948

** The first modern text on abelian varieties. In French.

*Henri Poincaré , "Analysis Situs", Journal de l'École Polytechnique ser 2,**1**(1895) pages 1-123.

* Henri Poincaré, "Complément à l'Analysis Situs", Rendiconti del Circolo matematico di Palermo,**13**(1899) pages 285-343.

* Henri Poincaré, "Second complément à l'Analysis Situs",Proceedings of the London Mathematical Society ,**32**(1900), pages 277-308.

* Henri Poincaré, "Sur certaines surfaces algébriques ; troisième complément à l'Analysis Situs", Bulletin de la Société mathématique de France,**30**(1902), pages 49-70.

* Henri Poincaré, "Sur les cycles des surfaces algébriques ; quatrième complément à l'Analysis Situs", Journal de mathématiques pures et appliquées, 5° série,**8**(1902), pages 169-214.

* Henri Poincaré, "Cinquième complément à l'analysis situs", Rendiconti del Circolo matematico di Palermo**18**(1904) pages 45-110.

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