Existence of a parallel vector field implies a splitting of the metric

Question

Where can I find a proof of the following claim:

Existence of a parallel vector field on a Riemannian manifold implies that the metric splits locally as a product of a one-dimensional manifold and $n-1$-dimensional one.

(By parallel I mean, parallel w.r.t the Levi-Civita connection).

I think that I can roughly see how it is done, but I am having trouble constructing a full proof. I guess the sectional curvature of any plane which contains the parallel vector should be zero, so we have all sorts of flat surfaces composing our manifold locally. But how can we create a "single split" which takes into account all of them?

Answer 1

I haven't been able to find a convenient reference. The result follows from the de Rham decomposition theorem [KN, Thm. 6.1], but that seems like overkill.

Here's a sketch of a simpler proof. Suppose $X$ is a parallel vector field on $M$, and $p$ is a point of $M$. By rescaling, we may assume $X$ is a unit vector field. Then we observe several things:

1. The fact that $X$ is parallel implies $d(X^\flat) = -2\text{Alt}(\nabla X^\flat) = 0$, so in a neighborhood $U$ of $p$ there is a smooth real-valued function $r$ such that $X = \operatorname{grad} r$ [IRM, Problem 5-13].
2. If we let $N=r^{-1}(0)$, then $|r|$ is equal to the geodesic distance to $N$ in $U$ (possibly after shrinking $U$) [IRM, Thm. 6.34].
3. We can choose Fermi coordinates $(x^1,\dots,x^n=r)$ on $U$ (again, possibly after shrinking further), and they satisfy $g_{nn}=1$, $g_{n\alpha} = g_{\alpha n} = 0$ for $\alpha = 1,\dots, n-1$, and $X = \partial/\partial x^n$ [IRM, Example 6.43 and Prop. 6.41].
4. The fact that $X$ is parallel implies that it is also a Killing vector field, so $\mathscr L_X (g) \equiv 0$. Expanding this out in coordinates shows that $\partial_{x^n} g_{\alpha\beta} \equiv 0$, so in fact $g$ has the following form in Fermi coordinates on $U$: $$ g = dr^2 + \sum_{\alpha,\beta=1}^{n-1} g_{\alpha\beta}(x^1,\dots,x^{n-1}) dx^\alpha dx^\beta. $$ This is a product metric on $(-\varepsilon,\varepsilon) \times N$.

References

• [IRM] John M. Lee, Introduction to Riemannian Manifolds, 2nd ed., Springer, 2018.
• [KN] Shoshichi Kobayashi & Katsumi Nomizu, Foundations of Differential Geometry, vol. 1, Wiley, 1996.