Basic Notions

  • A. L. Onishchik
Part of the Encyclopaedia of Mathematical Sciences book series (EMS, volume 20)


We will assume familiarity with the basic concepts of manifold theory. However in order to avoid misunderstandings some of them will be defined in the text. The basic field, by which we mean either the field ℝ of real numbers or the field ℂ of complex numbers, will be denoted by K. Unless stated otherwise, differentiability of functions will be understood in the following sense: in every case there exist as many derivatives as are needed. Differentiability of manifolds and maps is understood in the same sense. The Jacobian matrix of a system of differentiable functions f 1,…, f m of variables x 1,…, x n will be denoted by \(\frac{{\partial \left( {{{f}_{1}}, \ldots ,{{f}_{m}}} \right)}}{{\partial \left( {{{x}_{1}}, \ldots ,{{x}_{n}}} \right)}} \).For m = n its determinant will be denoted by \(\frac{{D\left( {{{f}_{1}}, \ldots ,{{f}_{n}}} \right)}}{{D\left( {{{x}_{1}}, \ldots ,{{x}_{n}}} \right)}} \)


Semidirect Product Differentiable Manifold Differentiable Structure Tangent Algebra Trivial Fibre Bundle 
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Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • A. L. Onishchik
    • 1
  1. 1.Yaroslavl UniversityYaroslavlRussia

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