Fold

A type of singularity of differentiable mappings (cf. Singularities of differentiable mappings).

Let $f:{\mathbf{R}}^n\to {\mathbf{R}}^n$ be a $C^{\mathrm{\infty }}$- function. Then $x_0\in {\mathbf{R}}^n$ is said to be a fold of $f$ if

$$\dim \mathrm{Ker}{f}^{\prime }(x_0)=\dim \mathrm{Coker}{f}^{\prime }(x_0)=1$$

and if the Hessian of $f$ at $x_0$ is not equal to zero (cf. Hessian of a function). This definition can be generalized to the case of a $C^{\mathrm{\infty }}$- mapping $f:X\to Y$ between $C^{\mathrm{\infty }}$- manifolds $X$ and $Y$( necessarily of the same dimension), cf. [a1].

The name derives from the following fact: If $f:X\to Y$( with $X$, $Y$ and $f$ as above) has a fold at $x_0\in X$, then there are local coordinates $(x_1\dots x_n)$ in $X$ vanishing at $x_0$ and local coordinates $(y_1\dots y_n)$ in $Y$ vanishing at $f(x_0)$ such that $f$ has the local representation

$$f(x_1\dots x_n)=(x_1\dots x_{n-}1,x_n^2).$$

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