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
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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
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References
[a1] | L.V. Hörmander, "The analysis of linear partial differential operators" , 3 , Springer (1985) |
[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) |
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