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.



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


(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?)



Nowhere vanishing, normalized vector field with bounded derivatives
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.


