A simple fact: Given a vector field on a compact manifold with boundary, if the vector field points inward along the boundary, then it must vanish somewhere in the interior. (EDIT: As pointed out in the accepted answer and in a comment, the Euler characteristic must be nonzero for this to be true.)


My question: Is there an analog of this fact in infinite dimensions? Perhaps for Banach manifolds?



vanishing of vector field in infinite dimensions
The simple fact in question is false in any dimension greater than one.


Consider the strip $ \mathbb{R} \times [-\pi/2,\pi/2] \subset \mathbb{R}^2$. At a point $(x, y)$ take the vector $(-sin(y), cos(y))$. This does not depend on $x$ so descends to a vector field on the annulus $\mathbb{R}/\mathbb{Z} \times [-\pi/2, \pi/2]$. It is obviously nowhere vanishing and points inwards at the boundaries. 


