TITLE: vanishing of vector field in infinite dimensions
QUESTION [6 upvotes]: 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?

REPLY [5 votes]: 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.