Ivancevic_Applied-Diff-Geom

Introduction 13the variables defined by other coordinate systems whenever charts overlap.This idea is often presented formally using transition maps.This allows one to extend the meaning of differentiability to spaces withoutglobal coordinate systems. Specifically, a differentiable structure allowsone to define a global differentiable tangent space, and consequently, differentiablefunctions, and differentiable tensor–fields (including vector–fields).Differentiable manifolds are very important in physics. Special kinds ofdifferentiable manifolds form the arena for physical theories such as classicalmechanics (Hamiltonian mechanics and Lagrangian mechanics), generalrelativity and Yang–Mills gauge theory. It is possible to develop calculuson differentiable manifolds, leading to such mathematical machinery as theexterior calculus.Historically, the development of differentiable manifolds (as well as differentialgeometry in general) is usually credited to C.F. Gauss and hisstudent B. Riemann. The work of physicists J.C. Maxwell and A. Einsteinlead to the development of the theory transformations between coordinatesystems which preserved the essential geometric properties. Eventuallythese ideas were generalized by H. Weyl in ‘Idee der Riemannschen Fläshe’(1913) and ‘Raum, Ziet, Materie’ (‘Space Time Matter’, 1921). T. Levi–Civita applied these ideas in ‘Lezioni di calcolo differenziale assoluto’ (‘TheAbsolute Differential Calculus’, 1923). The approach of Weyl was essentiallyto consider the coordinate functions in terms of other coordinatesand to assume differentiability for the coordinate function. In 1963, S.Kobayashi and K. Nomizu gave the group transformation/atlas approach.Generalizations of manifoldsThe three most common generalizations of manifolds are:• orbifolds: An orbifold is a generalization of manifold allowing for certainkinds of ‘singularities’ in the topology. Roughly speaking, it is aspace which locally looks like the quotients of some simple space (e.g.,Euclidean space) by the actions of various finite groups. The singularitiescorrespond to fixed points of the group actions, and the actionsmust be compatible in a certain sense.• algebraic varieties and schemes: An algebraic variety is glued togetherfrom affine algebraic varieties, which are zero sets of polynomials overalgebraically closed fields. Schemes are likewise glued together fromaffine schemes, which are a generalization of algebraic varieties. Both