Singularities of Mappings of Euclidean Spaces

  • Hassler Whitney
Part of the Contemporary Mathematicians book series (CM)


We shall describe here some results and methods pertaining to the following general problem (details will appear elsewhere). Suppose a mapping f 0of an open set R in n-spaceE n into m-space E m is given (we shall write f: R n E m ). How can we alter f 0slightly, obtaining a mapping f with nicer and simpler properties? By the Weierstrass approximation theorem (generalized), we may require that f be analytic in R; if f 0 was r-smooth (had continuous partial derivatives through the r th order), we may require the partial derivatives of f through the r th order to approximate those of f 0 (we then call f anr-approximation). Now take any regular point p of f, that is, a point p such that f is of maximum rank v = inf (n, m) at p. (Equivalently, using coordinate systems inE n and in E m , the Jacobian matrix of f at p is of rank v.) Then, by the implicit function theorem, we may choose coordinates so that f has the form
$${y^i} = {x^i}\left( {i = 1, \cdots ,v} \right), {y^i} = 0\left( {i > v, if m > n} \right).$$


Singular Point Null Space Smooth Manifold Stable Mapping Regular Point 
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© Birkhäuser Boston 1992

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  • Hassler Whitney

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