Дифеоморфізм

Дифеоморфі́зм — взаємно однозначне і неперервно диференційовне відображення $f:M\to N$ гладкого многовиду $M$ в гладкий многовид $N$ , обернене до якого теж є неперервно диференційовним. Зазвичай під гладкістю розуміють $\ C^{\infty }$ — гладкість, проте таким же чином можуть бути визначені дифеоморфізми з іншим типом гладкості, наприклад $\ C^{k}$ при будь-кому $k\in \mathbb {N}$ .

Пов'язані визначення

Якщо для $M$ та $N$ існує дифеоморфізм, то говорять, що $M$ й $N$ дифеоморфні. Множина дифеоморфізмів многовиду $M$ у собі утворює групу, що позначається $\operatorname {Diff} M$ .

Приклади

Нехай $\ f(x,y)=(x^{2}+y^{3},x^{2}-y^{3})$ . Матриця Якобі цього відображення дорівнює:

J_{f}=\left({\begin{array}{cc}2x&3y^{2}\\2x&-3y^{2}\end{array}}\right).

Її визначник дорівнює нулю тоді і тільки тоді коли $xy=0$ . Тобто f є дифеоморфізмом за межами x-осі і y-осі.

Література

Пришляк О.О.. Диференціальна геометрія : Курс лекцій. – К.: Видавничо-поліграфічний центр «Київський університет», 2004. – 68 с.
Милнор Дж., Уоллес А. Дифференциальная топология / Пер. с англ. — Москва: Мир, 1972. — 280 с.
Ф.Уорнер Основы теории гладких многообразий и групп — Москва: Мир, 1987. — 302 с.

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