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A Criterion Concerning Lie Groups

Differential Geometry
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Tsakanikas Nickos
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A Criterion Concerning Lie Groups

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Post by Tsakanikas Nickos » Thu Jun 09, 2016 10:16 pm

Definition: A smooth manifold $G$ with a group structure such that the multiplication map $ \cdot \ \colon G \times G \longrightarrow G \ , \ (g,h) \mapsto g \cdot h$ and the inversion map $ G \longrightarrow G \ , \ g \mapsto g^{-1} $ are smooth is called a Lie Group.



Let $G$ be a smooth manifold with a group structure such that the map $ G \times G \longrightarrow G \ , \ (g,h) \mapsto gh^{-1} $ is smooth. Show that $G$ is a Lie Group.
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