The real projective plane ${\bf P}^2({\bf R})$ is the only closed surface with the fixed point property: all continuous maps from the plane to itself has a fixed point. The projective spaces ${\bf P}^{2k}({\bf R})$ and ${\bf P}^{2k}({\bf C})$ also have this property; this follows from the Lefschetz fixed point formula. There seems to be few examples of manifolds with this property though. I think that odd dimensional orientable manifolds always have a vector field without zeros, and a map without fixed points can be obtained by integrating such a vector field.

Are there any other examples of compact manifolds without boundary with the fixed point property? Please explain briefly how the fixed point property is obtained.

Is there some hope to get a complete list of such manifolds?