Differential Variety

A topological variety $M$ of dimension $n$ is a topological Espace separate, such as each one of its points admits a Voisinage open homeomorphic to open of Euclidean space $\ mathbb {R} ^n$ . More precisely:

$\ forall X \ in M$ , there exists an open vicinity $U_x$ and a Homéomorphisme $\ varphi_x: U_x \ longrightarrow \ varphi_x (U_x) \ subset \ mathbb {R} ^n$ . It is said whereas $(U_x, \ varphi_x)$ is a local chart of $M$ .

Charts $(U_x, \ varphi_x)$ which recover (entirely) $M$ constitute a atlas $M$ variety.

Being given such an atlas, one says that $M$ is a differential variety of class $\ mathcal {C} ^k, \ K \ Ge 1$ for this atlas if:

$\ forall \ X, \ y$ , such as $U_x \ course U_y \ neq \ varnothing$ , then $\ varphi_x \ circ \ varphi_y^ {- 1}$ is a Difféomorphisme of class $\ mathcal {C} ^k$ .

Properties

The differential varieties form a category whose morphisms are the differentiable applications

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