Ivancevic_Applied-Diff-Geom

Introduction 5• Riemannian manifolds, on which distances and angles can be defined;• symplectic manifolds, which serve as the phase space in mechanicsand physics;• 4D pseudo–Riemannian manifolds which model space–time in generalrelativity.The study of manifolds combines many important areas of mathematics:it generalizes concepts such as curves and surfaces as well as ideas fromlinear algebra and topology. Certain special classes of manifolds also haveadditional algebraic structure; they may behave like groups, for instance.Historically, before the modern concept of a manifold there were severalimportant results:(1) Carl Friedrich Gauss was arguably the first to consider abstract spacesas mathematical objects in their own right. His ‘Theorema Egregium’gives a method for computing the curvature of a surface S withoutconsidering the ambient Euclidean 3D space R 3 in which the surfacelies. Such a surface would, in modern terminology, be called a manifold.(2) Non–Euclidean geometry considers spaces where Euclid’s ‘Parallel Postulate’fails. Saccheri first studied them in 1733. Lobachevsky, Bolyai,and Riemann developed them 100 years later. Their research uncoveredtwo more types of spaces whose geometric structures differ from thatof classical Euclidean nD space R n ; these gave rise to hyperbolic geometryand elliptic geometry. In the modern theory of manifolds, thesenotions correspond to manifolds with negative and positive curvature,respectively.(3) The Euler characteristic is an example of a topological property (or topologicalinvariant) of a manifold. For a convex polyhedron in Euclidean3D space R 3 , with V vertices, E edges and F faces, Euler showed thatV − E + F = 2. Thus the number 2 is called the Euler characteristicof the space R 3 . The Euler characteristic of other 3D spaces is a usefultopological invariant, which can be extended to higher dimensions usingthe so–called Betti numbers. The study of other topological invariantsof manifolds is one of the central themes of topology.(4) Bernhard Riemann was the first to do extensive work generalizing theidea of a surface to higher dimensions. The name manifold comes fromRiemann’s original German term, ‘Mannigfaltigkeit’, which W.K. Cliffordtranslated as ‘manifoldness’. In his famous Göttingen inaugural