Godbillon-Vey invariant

In mathematics , the Godbillon-Vey invariant is an invariant of scrolls .

definition

Let be a smooth, transversally orientable, -dimensional foliation of a -dimensional manifold . Their tangential hyperplane field can be (locally) as a zero set of a - form ${\displaystyle {\mathcal {F}}}$ ${\displaystyle (n-q)}$ ${\displaystyle n}$ ${\displaystyle M}$ ${\displaystyle E\subset TM}$ ${\displaystyle q}$

${\displaystyle \omega \in \Omega ^{q}(M)}$

and there is (locally) a form with ${\displaystyle 1}$ ${\displaystyle \eta \in \Omega ^{1}(M)}$

${\displaystyle d\omega =\omega \wedge \eta }$.

The Godbillon-Vey invariant of foliation is defined as ${\displaystyle {\mathcal {F}}}$

${\displaystyle gv({\mathcal {F}}):=\int _{M}\eta \wedge (d\eta )^{q}}$.

The definition is independent of the choice of and . ${\displaystyle \omega }$ ${\displaystyle \eta }$

Duminy's theorem

A leaf of a scroll is called resilient if it is not actually embedded and has nontrivial holonomy . ${\displaystyle L}$ ${\displaystyle {\mathcal {F}}}$

The Godbillon-Vey invariant of Kodimension 1 foliage measures the resilience of leaves in the following sense.

Duminy's theorem : Let be a smooth, transversely orientable, -dimensional foliation of a -dimensional manifold . If no leaf is resilient, then it is ${\displaystyle {\mathcal {F}}}$ ${\displaystyle (n-1)}$ ${\displaystyle n}$ ${\displaystyle M}$ ${\displaystyle {\mathcal {F}}}$

${\displaystyle gv({\mathcal {F}})=0}$.

literature

• A. Candel and L. Conlon, Foliations. I, American Mathematical Society, Providence, RI, 2000.
• Godbillon, Vey Un invariant des feuilletages de codimension 1 , Compte Rendu Academie des Sciences, Paris, Volume 273, 1971, pp. 273-292
• Étienne Ghys L'invariant de Godbillon-Vey , Séminaire Bourbaki 706, 1988/89