Razdelyayushchie resheniya i otnoshenie poryadka
(Prilozhenie 3 iz knigi "Malochleny")

A.G.Khovanskii

My budem rassmatrivat' raspredeleniya s osobennostyami koorientirovannykh
giperploskostei na mnogoobrazii $M^n$. Po opredeleniyu takoe raspredelenie
zadaetsya uravneniem Pfaffa $\alpha =0$, v kotorom 1-forma $\alpha $
opredelena s tochnost'yu do umnozheniya na polozhitel'nuyu funktsiyu.
Mnozhestvom $O$ osobykh tochek raspredeleniya nazyvaetsya mnozhestvo
tochek, v kotorykh forma $\alpha $ obrashchaetsya v tozhdestvennyi nol'.

Opredelenie. Mnozhestvo $A\subset M^n\setminus O$ obladaet svoistvom
Rollya dlya raspredeleniya $\alpha =0$, esli dlya vsyakoi gladkoi krivoi
$\gamma \:[0,1]\to M^n$, kontsy $\gamma (0)$, $\gamma (1)$ kotoroi lezhat
v mnozhestve $A$, a vektora skorosti v nachal'nyi i konechnyi moment
transversal'ny raspredeleniyu, to est' $\alpha (\dot\gamma (0))\neq 0$ i
$\alpha (\dot\gamma (1))\neq 0$, naidetsya tochka kontakta s
raspredeleniem, to est' naidetsya chislo $\xi $, $0\leq \xi\leq 1$, takoe,
chto $\alpha (\dot\gamma (\xi))=0$.

Osnovnym rezul'tatom nastoyashchego prilozheniya yavlyaetsya
sleduyushchaya

Teorema 1. Mnozhestvo $A$ obladaet svoistvom Rollya otnositel'no
raspredeleniya $\alpha =0$, esli i tol'ko esli na mnogoobrazii
$M^n\setminus O$ sushchestvuet razdelyayushchee reshenie uravneniya
$\alpha =0$, soderzhashchee mnozhestvo $A$.