Nowhere vanishing, normalized vector field with bounded derivatives

Question

It is well-known that any non-compact manifold admits a nowhere vanishing vector field. If we have a Riemannian metric we may pick such a vector field and normalize it so that at every point it has unit length. But what I additionally need is a control on the derivatives of the vector field and here it seems to me that the usual constructions of non-vanishing vector fields break down.

Let M be a non-compact manifold of bounded geometry (i.e., the curvature tensor and all its derivatives are bounded, and the injectivity radius is uniformly positive).

Does there exist a nowhere vanishing, normalized vector field X on M, such that $\| \nabla^k X \|_\infty < C_k$ for all $k \in \mathbb{N}$?

(If yes, may we even drop the assumption about having bounded geometry? If no, what if we weaken the requirement to $\|\nabla X\|_\infty < C$, i.e., just the first derivative being bounded?)

Answer 1

I could find now a solution in the literature. In

S. Weinberger, Fixed-point theories on noncompact manifolds, J. Fixed Point Th. Appl. 6 (2009), 15-25

it is shown that a normed vector field with bounded derivatives exists if and only if the Euler class $e(M) \in H_0^{\mathrm{uf}}(M)$ vanishes.