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TOWARDS DIFFERENTIAL CALCULUS IN STRATIFIED GROUPS

Part of: Functions of several variables Lie groups

Published online by Cambridge University Press:  17 June 2013

VALENTINO MAGNANI
VALENTINO MAGNANI*
Affiliation:
Dipartimento di Matematica, Università di Pisa, Largo Bruno Pontecorvo 5, I-56127, Pisa email magnani@dm.unipi.it
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Abstract

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We 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.

Keywords

stratified groupsimplicit function theoremmean value inequalityrank theoremintrinsic submanifolds

MSC classification

Secondary: 22E30: Analysis on real and complex Lie groups 26B10: Implicit function theorems, Jacobians, transformations with several variables 26B12: Calculus of vector functions
Type
Research Article
Information
Journal of the Australian Mathematical Society , Volume 95 , Issue 1 , August 2013 , pp. 76 - 128
Copyright
Copyright ©2013 Australian Mathematical Publishing Association Inc. 

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