Differentiable Manifolds

  • E. Hlawka
Conference paper
Part of the Acta Physica Austriaca book series (FEWBODY, volume 7/1970)


Let M be a set. An n-dimensional chart c on M is a pair (U,ϕ) , where U is a subset of M and ϕ a bijective mapping of U onto an open set of Rn . U is called the domain of c,ϕ the coordinate mapping of c. For each pεU we can write (see Fig. 1)
$${\rm{f(p)}}\,{\rm{ = }}\,{\rm{(}}{{\rm{f}}_{{\rm{1}}\,}}{\rm{(p)}}\,{\rm{,}}...{\rm{,}}{{\rm{f}}_{\rm{n}}}\,{\rm{(p)}}\,{\rm{)}}\,{\rm{ = }}\,{\rm{(}}{{\rm{x}}_{\rm{1}}}\,{\rm{,}}...{\rm{,}}{{\rm{x}}_{\rm{n}}})$$
We call (x1,...,xn) the local coordinates of p in the chart (U,ϕ). The notation can be made more explicit by writing x1,...,xn in the form
or, in short, x(p)


Vector Field Coordinate Function Tensor Field Leibniz Rule Differentiable Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


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Copyright information

© Springer-Verlag 1970

Authors and Affiliations

  • E. Hlawka
    • 1
  1. 1.Mathematisches InstitutUniversität WienAustria

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