scholarly journals A proof of A. Gabrielov’s rank theorem

2021 ◽  
Vol 8 ◽  
pp. 1329-1396
Author(s):  
André Belotto da Silva ◽  
Octave Curmi ◽  
Guillaume Rond
Keyword(s):  
Rank Theorem

scholarly journals TOWARDS DIFFERENTIAL CALCULUS IN STRATIFIED GROUPS

2013 ◽  
Vol 95 (1) ◽  
pp. 76-128 ◽  
Author(s):  
VALENTINO MAGNANI
Keyword(s):  
Implicit Function Theorem ◽  
Group Structure ◽  
Mean Value ◽  
Implicit Function ◽  
Inverse Mapping ◽  
Natural Group ◽  
Stratified Groups ◽  
First Order ◽  
Inverse Mapping Theorem ◽  
Rank Theorem

AbstractWe study graded group-valued continuously differentiable mappings defined on stratified groups, where differentiability is understood with respect to the group structure. We characterize these mappings by a system of nonlinear first-order PDEs, establishing a quantitative estimate for their difference quotient. This provides us with a mean value estimate that allows us to prove both the inverse mapping theorem and the implicit function theorem. The latter theorem also relies on the fact that the differential admits a proper factorization of the domain into a suitable inner semidirect product. When this splitting property of the differential holds in the target group, then the inverse mapping theorem leads us to the rank theorem. Both implicit function theorem and rank theorem naturally introduce the classes of image sets and level sets. For commutative groups, these two classes of sets coincide and correspond to the usual submanifolds. In noncommutative groups, we have two distinct classes of intrinsic submanifolds. They constitute the so-called intrinsic graphs, that are defined with respect to the algebraic splitting and everywhere possess a unique metric tangent cone equipped with a natural group structure.

scholarly journals A rank theorem for analytic maps between power series spaces

1994 ◽  
Vol 80 (1) ◽  
pp. 95-115 ◽  
Author(s):  
Herwig Hauser ◽  
Gerd Müller
Keyword(s):  
Power Series ◽  
Rank Theorem ◽  
Analytic Maps
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