Real projective space

Definition

Real projective space of dimension $n$, denoted as $\mathbb {R} \mathbb {P} ^{n}$, is defined as the quotient of the set $\mathbb {R} ^{n+1}\setminus \{0\}$ via the group action of $\mathbb {R} ^{*}$ by scalar multiplication. Equivalently, it is the set of all directions in $\mathbb {R} ^{n+1}$, suitably topologized.

It can also be viewed as the quotient of the $n$-dimensional sphere via identification of antipodes. This is because $S^{n}$ (the $n$-sphere is the quotient of $\mathbb {R} ^{n+1}\setminus \{0\}$ under the action of the multiplicative group of positive real numbers, which is a normal subgroup of index two in the multiplicative group of all nonzero real numbers.

Geometric properties

Metric structure

The real projective space can be viewed as a Riemannian manifold when equipped with the Fubini-Study metric.

Curvature

The Fubini-Study metric is a constant-curvature metric. This constant curvature is negative for even dimensions and positive for odd dimensions.

Automorphisms and symmetries

Homogeneousness

Real projective spaces are homogeneous, in the sense that given any two points in the real projective space, there is a Riemannian isometry taking one to the other.