Editing Hairy ball theorem (section)

==Corollary==
A consequence of the hairy ball theorem is that any continuous [[Functions (mathematics)|function]] that maps an even-dimensional sphere [[Endomorphism|into itself]] has either a [[fixed point (mathematics)|fixed point]] or a point that maps onto its own [[antipodal point]].  This can be seen by transforming the function into a tangential vector field as follows.

Let ''s'' be the function mapping the sphere to itself, and let ''v'' be the tangential vector function to be constructed.  For each point ''p'', construct the [[stereographic projection]] of ''s''(''p'') with ''p'' as the point of tangency.  Then ''v''(''p'') is the [[displacement vector]] of this projected point relative to ''p''.  According to the hairy ball theorem, there is a ''p'' such that ''v''(''p'') = '''0''', so that ''s''(''p'') = ''p''.

This argument breaks down only if there exists a point ''p'' for which ''s''(''p'') is the antipodal point of ''p'', since such a point is the only one that cannot be stereographically projected onto the tangent plane of ''p''.

A further corollary is that any even-dimensional [[Real projective space|projective space]] has the [[fixed-point property]]. This follows from the previous result by [[Covering space#Lifting property|lifting]] continuous functions of <math>\mathbb{RP}^{2n}</math> into itself to functions of <math>S^{2n}</math> into itself.