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Transversal volumes of foliations definable in \(o\)-minimal structures. (Volumes transverses aux feuilletages définissables dans des structures \(o\)-minimales.) (French) Zbl 1076.53026

The main result of the paper is as follows: Let \({\mathcal F}\) be a foliation of codimension \(p\), with closed leaves, on the \({\mathcal C}^2\) submanifold \(M\subset {\mathbb R}^n\), and let \(X\) be a compact subset of \(M\). Let \(\Omega\) be a differential form of degree \(p\) on \(M\), such that \(i_Z\Omega=0\) for every vector field \(Z\) tangent to \({\mathcal F}\). Let us suppose that \(M\), \(X\), \({\mathcal F}\) and \(\Omega\) are \({\mathcal A}\)-definable objects, for some given \(o\)-minimal structure \({\mathcal A}\) on \({\mathbb R}^n\) [see L. van den Dries and C. Miller, Duke Math. J. 84, No. 2, 497–540 (1996; Zbl 0889.03025)]. Then there exists a definable subset \(\Gamma\subset X\), of dimension \(p\), which satisfies the following property: if \(C\subset X\) is a submanifold of dimension \(p\) such that \[ \int_C{| \Omega |} > \int_\Gamma{| \Omega |} \] then \(C\) cuts some leaf of \({\mathcal F}\) in at least two points.

MSC:

53C12 Foliations (differential geometric aspects)
49Q15 Geometric measure and integration theory, integral and normal currents in optimization
03C64 Model theory of ordered structures; o-minimality

Citations:

Zbl 0889.03025
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