Abstract: It is an open problem to determine when an action of R^k on a given manifold has a fixed point.
When k=1 the index theory of Poincaré-Hopf give a tool to localize zeros of vector fields. Such a theory does
not exist in higher dimension. In dimension 2, Lima proved that an action of R^k on a closes surface of
nonzero Euler characteristic always has a fixed point.
I will speak about recent work in the 3-dimensional case. This is joint work with C.Bonatti and B.Santiago.