Fold
From Encyclopedia of Mathematics
A type of singularity of differentiable mappings (cf. Singularities of differentiable mappings).
Let 
be a 
-function. Then 
is said to be a fold of 
if
 |
and if the Hessian of 
at 
is not equal to zero (cf. Hessian of a function). This definition can be generalized to the case of a 
-mapping 
between 
-manifolds 
and 
(necessarily of the same dimension), cf. [a1].
The name derives from the following fact: If 
(with 
, 
and 
as above) has a fold at 
, then there are local coordinates 
in 
vanishing at 
and local coordinates 
in 
vanishing at 
such that 
has the local representation
 |
References
| [a1] | L.V. Hörmander, "The analysis of linear partial differential operators" , 3 , Springer (1985) MR1540773 MR0781537 MR0781536 Zbl 0612.35001 Zbl 0601.35001 |
| [a2] | V.I. Arnol'd, S.M. [S.M. Khusein-Zade] Gusein-Zade, A.N. Varchenko, "Singularities of differentiable maps" , 1 , Birkhäuser (1985) (Translated from Russian) MR777682 Zbl 0554.58001 |
How to Cite This Entry:
Fold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fold&oldid=13658
Fold. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Fold&oldid=13658