Abstract
We establish a sufficient condition for injectivity in a class of mappings defined on open connected subsets of Rn , for arbitrary n. The result relates solvability of the appropriate vector fields with injectivity of the mapping and extends a result proved by the first author for n < 3 . Furthermore, we extend the result to connected paracompact smooth oriented manifolds and show that the convexity condition imposes strong topological restrictions on the manifold.
Highlights
Let be an open connected subset of Rn, and consider ( ) = {F ∈ C∞(, Rn) : det(DF)(x) = 0 ∀x}
As in Santos Filho, for each i ∈ {1, 2, . . . , n}, we consider VF,i the real vector field defined by VF,i (φ)(x) = det(DFi,φ)(x)), for all φ ∈ C∞
The mapping Fi,φ is given by: Its j-component is equal to f j if j = i and its i-component is equal to φ
Summary
DOS SANTOS FILHO1 and JOAQUIM TAVARES2 1Departamento de Matemática, Universidade Federal de São Carlos Via Washington Luis, km 235, 13565-905 São Carlos, SP, Brasil 2Departamento de Matemática, Universidade Federal de Pernambuco Av. Prof. Luiz Freire s/n, 50740-540 Recife, PE, Brasil. Manuscript received on November 3, 2008; accepted for publication on April 14, 2010
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