Let be a map of degree zero. Show that there exists points with and . Use this to show that if F is a continuous vector field defined on the unit ball in such that for all x, then there exits a point in where F points radially outward and another point in where F points radially inward.

Solution:

Suppose g is the antipodal map.

If for all then . This implies hence giving us a contradiction.

Similarly if then for all . This implies is homotopic to the antipodal map. But then . This is impossible as . Hence we again have a contradiction.

Now define . It is clearly a continuous vector field from . (It is well defined as F(x) never vanishes. Also restriction of F to is continuous as F is continuous. Finally as F is continuous, composing it with absolute value function is continuous. is continuous and F is continuous implies their ratio is continuous).

Note that where the first map is inclusion and the second map is . Since (as n-disk is contractible), therefore implying is of zero degree.

As is of degree zero therefore there is a point such that . Hence . This F points radially outward at this point (c is positive constant).

Similarly there is a point such that . This is the point where F points radially inward.