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Lie Groups are Groups that are additionally smooth manifolds such that the multiplication and the inverse maps are smooth.
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Left invariant connection on Lie groups
Let $M^n$ be a Riemannian manifold, $G^{n + k}, \: k\geq 1 $ be a connected Lie group equipped with a left invariant metric and $f:M^n \to G^{n + k}$ one isometric immersion. Let $X, Y$ left invariant …