TITLE: Rational homology spheres and geometric properties of the Wu manifold
QUESTION [6 upvotes]: I am interested in simply connected rational homology spheres. The first such example is in dimension 5 and it is the Wu manifold $SU(3)/SO(3)$. You can find a discussion about simply connected rational spheres in: 
Simply-connected rational homology spheres.
I need to understand the geometric properties of this manifold in order to construct mappings with certain properties. So my questions are:
1. Why is the Wu manifold important except for being a simply connected rational homology sphere? 
2. Where can I read about the geometric properties of the Wu manifold?
I am interested more in geometric properties of $SU(3)/SO(3)$ rather than its applications to algebraic topology since, it seems, I have to construct certain mappings explicitly without using algebraic topology. I will describe my problem later in another post.

REPLY [6 votes]: Another feature of the Wu manifold which makes it an important example is that it is not spin${}^c$. In contrast, every closed orientable manifold of dimension at most four is spin${}^c$.
One geometric property of the Wu manifold is that it admits a metric of positive scalar curvature. Gromov and Lawson showed that a simply connected non-spin manifold of dimension at least five always admits a metric of positive scalar curvature.