Afinní konexe

Afinní konexe je geometrický objekt na hladké varietě, který spojuje okolní tečné prostory. Pojem afinní konexe má své kořeny v geometrii 19. století a tenzorových počtech, ale nebyl plně rozvinutý až do roku 1920, kdy jej popsali Élie Cartan (jako součást jeho obecné teorie konexí) a Hermann Weyl (který používal tento pojem, jako součást jeho základů pro obecnou teorii relativity).

Definice

Nechť M je hladká varieta a C^∞(M,TM) je prostor vektorových polí na M, která je prostorem hladké sekce tečného bandlu TM. Potom afinní konexe na M je a bilineární zobrazení:

{\begin{matrix}C^{\infty }(M,TM)\times C^{\infty }(M,TM)&\rightarrow &C^{\infty }(M,TM)\\(X,Y)&\mapsto &\nabla _{X}Y,\end{matrix}}

taková, že pro všechny hladké funkce f v C^∞(M,R) a všechna vektorová pole X, Y na M:

$\nabla _{fX}Y=f\nabla _{X}Y$ , to znamená, že $\nabla$ je C^∞(M,R)-lineární v první proměnné;
$\nabla _{X}(fY)=\mathrm {d} f(X)Y+f\nabla _{X}Y$ , to znamená, že $\nabla$ splňuje Leibnizovo pravidlo ve druhé proměnné.

Pahýl

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