# 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 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

• ''such that in which $p \in M$ are'' means ''such that at every point $p \in M$ they are''? Sep 30 '14 at 10:17
• That's right Ben McKay. Such that at every point $p \in M$ they are tangents to $M$
– user58848
Sep 30 '14 at 22:45
• @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. Jan 29 '16 at 10:23