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==Definition==

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

It can also be viewed as the quotient of the <math>n</math>-dimensional [[sphere]] via identification of [[antipode]]s. This is because <math>S^n</math> (the <math>n</math>-sphere is the quotient of <math>\R^{n+1}\setminus \{ 0 \}</math> 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 space|homogeneous]], in the sense that given any two points in the real projective space, there is a Riemannian isometry taking one to the other.

Real projective space - Revision history

Vipul: 1 revision

Vipul at 13:29, 18 June 2007