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)?
