# Orthogonal distribution to parallel vector fields

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...

## 1 Answer

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.

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