Geometry of Real Projective Spaces
Real projective space admits a constant positive scalar curvature metric, coming from the double cover by the standard round sphere (the antipodal map is locally an isometry).
For the standard round metric, this has sectional curvature identically 1.
In the standard round metric, the measure of projective space is exactly half the measure of the sphere.
Read more about this topic: Real Projective Space
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