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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 vector fields in $G$ such that in which $ p \in M $ are tangent to $ M $. Does anyone know any non trivial example where $\overline{\nabla}_{X} Y$ is also tangent to $M$ at $p$?

Here $\overline{\nabla} $ is the Riemannian connection on $G$, that for left invariant vector fields $X, Y$ is given by $$\overline{\nabla}_X Y = \frac{1}{2}\{[X, Y] - ad_{X}^{*}Y - ad_{Y}^{*}X \}$$ Thanks

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  • $\begingroup$ ''such that in which $p \in M$ are'' means ''such that at every point $p \in M$ they are''? $\endgroup$
    – Ben McKay
    Commented Sep 30, 2014 at 10:17
  • $\begingroup$ That's right Ben McKay. Such that at every point $p \in M$ they are tangents to $M$ $\endgroup$
    – user58848
    Commented Sep 30, 2014 at 22:45
  • $\begingroup$ @Jao: I do not know if you are still interested but here is a possible answer. Take as $M$ a subgroup $H$ whose action is right-invariant (see mathoverflow.net/questions/229268/…). Then for $X,Y$ in the Lie algebra of $H$ you have $\overline{\nabla}_XY$ is always tangent to $M$ since $M$ is totally geodesic. $\endgroup$
    – Holonomia
    Commented Jan 29, 2016 at 10:23

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