# Real projective space

## Contents

## Definition

**Real projective space** of dimension , denoted as , is defined as the quotient of the set via the group action of by scalar multiplication. Equivalently, it is the set of all *directions* in , suitably topologized.

It can also be viewed as the quotient of the -dimensional sphere via identification of antipodes. This is because (the -sphere is the quotient of 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.