synthetic differential geometry
Introductions
from point-set topology to differentiable manifolds
geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry
Differentials
Tangency
The magic algebraic facts
Theorems
Axiomatics
Models
differential equations, variational calculus
Chern-Weil theory, ∞-Chern-Weil theory
Cartan geometry (super, higher)
A smooth map between smooth manifolds is called stable if every nearby smooth function, hence every function in some open neighbourhood of inside the mapping space , is equal to up to conjugation with a pair of diffeomorphisms.
Similarly, is infinitesimally stable if this statement holds to first order in derivatives, in a suitable sense.
(due to Mather 68, see Ruas 22, Def. 2.2, Def. 3.3).
A proper smooth map between smooth manifolds is “stable” (Def. ) if and only if it is infinitesimally stable.
(Mather 70, Thm. 4.1, Ruas 22, Thm. 3.11)
The original articles:
John N. Mather, Stability of mappings: I. The division theorem, Annals of Mathematics 87 1 (1968) 89 doi:10.2307/1970595
John N. Mather, Stability of Mappings: II. Infinitesimal Stability Implies Stability, Annals of Mathematics 89 2 (1969) 254 doi:10.2307/1970668
John N. Mather, Stability of mappings: V. Transversality, Advances in Mathematics 4 3 (1970) 301-336 (doi:10.1016/0001-8708(70)90028-9)
Review:
A proof in synthetic differential topology is provided in section 7.3 of
following
Last revised on May 17, 2022 at 13:29:50. See the history of this page for a list of all contributions to it.