Fold

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

Let $ f : \mathbf R ^ {n} \rightarrow \mathbf R ^ {n} $ be a $ C ^ \infty $- function. Then $ x _ {0} \in \mathbf R ^ {n} $ is said to be a fold of $ f $ if

$$ { \mathop{\rm dim} \mathop{\rm Ker} } f ^ { \prime } ( x _ {0} ) = \ { \mathop{\rm dim} \mathop{\rm Coker} } f ^ { \prime } ( x _ {0} ) = 1 $$

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 ^ \infty $- mapping $ f : X \rightarrow Y $ between $ C ^ \infty $- manifolds $ X $ and $ Y $( necessarily of the same dimension), cf. [a1].

The name derives from the following fact: If $ f : X \rightarrow 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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