# Orthonormal vector fields on a Riemannian manifold

Let $$(M,g)$$ be a Riemannian manifold. We equip the tangent bundle $$TM$$ with the Sasaki metric $$g_s$$. Assume that $$X: M \to TM$$ is a vector field on $$M$$.

We say that $$X$$ is an orthonormal vector field if $$X^* g_s=g$$. We say that $$X$$ is a generalized orthonormal vector field if $$X^* g_s=\lambda g$$ for some constant $$\lambda$$. The motivation for such a terminology is that an orthonormal matrix $$A\in M_n(\mathbb{R})$$ defines a vector field $$A:\mathbb{R}^n\to \mathbb{R}^n\times \mathbb{R}^n =T\mathbb{R}^n$$ which pulls back the satandard metrics, that is the Sasaki metric associated to the Euclidean metric, to the original Euclidean metric of $$M=\mathbb{R}^n$$.

Questions: Let $$(M,g)$$ be a compact Riemannian manifold whose Euler characteristic is zero. Does $$M$$ admit a non vanishing orthonormal vector field?

The second question:

Let $$(M,g)$$ be an arbitrary Riemannian manifold with a (generalized) orthonormal vector field $$X$$ on $$M$$ with singular set $$X$$. Is the restriction of the flow of $$X$$ to $$M\setminus S$$ a geodesible flow?(A flow whose trajectories are unparametrized geodesics for a Riemannian metric)?

• The first that comes to mind is the observation that $X$ is orthonormal if and only if for any $p \in M$ the differential of $X$ (considered as a map $M \to TM$) at $p$ sends $T_p M$ to the horizontal subspace $H_{(p, X_p)} \subseteq T_{(p, X_p)} (TM)$ (one just needs to unravel the definition of the Sasaki metric). But then it follows that $dX_p \colon T_p M \to H_{(p, X_p)}$ is an isometric isomorphism. – Ivan Solonenko Dec 10 '18 at 12:21