Атлас (математика)

Атлас — поняття диференціальної геометрії, що дозволяють ввести гладку структуру на многовиді .

Визначення

Нехай $K$ — числове поле (наприклад $\mathbb {R}$ або $\mathbb {C}$ ), $X$ — топологічний простір.

Карта — це пара $(U,f)$ , де

U

— відкрита множина в

X

f

— гомеоморфізм з

U

у відкриту множину в

K^{n}

Якщо області визначення двох карт $\,(U_{1},f_{1})$ і $\,(U_{2},f_{2})$ перетинаються ( $U_{1}\cap U_{2}\neq \emptyset$ ), то між множинами $f_{1}^{-1}(U_{2})$ і $f_{2}^{-1}(U_{1})$ є взаємно обернені відображення (гомоморфізми), що називаються відображеннями склеюваннями :
${\begin{matrix}f_{12}=f_{1}\circ f_{2}^{-1}|_{f_{2}(U_{1}\cap U_{2})}&:\ f_{2}(U_{1}\cap U_{2})\to f_{1}(U_{1}\cap U_{2})\\f_{21}=f_{2}\circ f_{1}^{-1}|_{f_{1}(U_{1}\cap U_{2})}&:\ f_{1}(U_{1}\cap U_{2})\to f_{2}(U_{1}\cap U_{2})\end{matrix}}$

Атлас — це множина узгоджених карт $\,\{(U_{\alpha },f_{\alpha })\}$ , $\alpha \in {\mathcal {A}}$ , така, що $\,\{U_{\alpha }\}$ утворює покриття простору $X$ . Тут ${\mathcal {A}}$ — деяка множина індексів. При цьому атлас називається гладким (класу $\,C^{k}$ ) або аналітичним, якщо функції заміни координат $\,f_{\alpha _{1}\,\alpha _{2}}$ для всіх карт гладкі (класу $\,C^{k}$ ) або аналітичні.

Два гладкі (аналітичні) атласи називаються узгодженими, якщо їх об'єднання також є гладким (аналітичним) атласом.

Література

Lee, John M. (2006). Introduction to Smooth Manifolds. Springer-Verlag. ISBN 978-0-387-95448-6.
Sepanski, Mark R. (2007). Compact Lie Groups. Springer-Verlag. ISBN 978-0-387-30263-8.

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