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Let $(M,g)$ be a riemannian manifold. A vector field $X$ on $M$ is called parallel if the Levi-Civita connection $D_YX$ vanishes for every choice of $Y$. I already know that given $k$ parallel vector fields $X_1,...,X_k$ the distribution $\mathcal{D}$ spanned by these vector fields is integrable. I'm trying to prove that the orthogonal distribution $\mathcal{D}^\bot$ is integrable.

My attempt

Clearly, given a point $m\in M$: $$\mathcal{D}^\bot_m=(X_1|_m)^\bot \cap ... \cap (X_k|_m)^\bot.$$ So, since the intersection of integrable distributions is integrable, I just need to prove the $k=1$ case: $$\mathcal{D}^\bot_m=(X|_m)^\bot.$$ Basically, by Frobenius theorem, I have to prove that given two vector fields $V,W\bot X$, then $[V,W]\bot X$. My idea was to use the compatibility of Levi-Civita connection with the metric to show that $g([V,W],X)$ has to be constant: $$U(g([V,W],X))=g(D_U[V,W],X)$$ and here I'm stuck...

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1 Answer 1

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Hint: Try using torsion-freeness to write $[V,W]=D_VW-D_WV$ and then exploit compatibility with the metric to analyze $g([V,W],X)$ directly.

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  • $\begingroup$ Thank you very much. Lately, I've been giving up too early and I don't know why... $\endgroup$
    – Kandinskij
    May 8, 2023 at 13:15
  • $\begingroup$ You're welcome. For this exercise, things get tricky if you don't have the right idea how to start ... $\endgroup$ May 9, 2023 at 6:55

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