Quaternionic projective space

In <a href="/facts/Mathematics/pxTouaz4">mathematics</a>, quaternionic projective space is an extension of the ideas of <a href="/facts/Real_projective_space/svbABDtd">real projective space</a> and <a href="/facts/Complex_projective_space/QmgneIzS">complex projective space</a>, to the case where coordinates lie in the ring of <a href="/facts/Quaternion/T3tnu7mX">quaternions</a> 
 
 
 
 
 H
 
 .
 
 
 {\displaystyle \mathbb {H} .}
 
 Quaternionic projective space of dimension n is usually denoted by

H
            P
          
          
            n
          
        
      
    
    {\displaystyle \mathbb {HP} ^{n}}

and is a <a href="/facts/Closed_manifold/jMG6G6Bl">closed manifold</a> of (real) dimension 4n. It is a <a href="/facts/Homogeneous_space/YGebqcJA">homogeneous space</a> for a <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> action, in more than one way. The quaternionic projective line 
 
 
 
 
 
 H
 P
 
 
 1
 
 
 
 
 {\displaystyle \mathbb {HP} ^{1}}
 
 is homeomorphic to the 4-sphere.

Quaternionic projective space open-in-new

Quaternionic projective space