  A gauge-fixing procedure for the Yang-Mills theory on an n-dimensional sphere
(or a hypersphere) is discussed in a systematic manner. We claim that Adler's
gauge-fixing condition used in massless Euclidean QED on a hypersphere is not
conventional because of the presence of an extra free index, and hence is
unfavorable for the gauge-fixing procedure based on the BRST invariance
principle (or simply BRST gauge-fixing procedure). Choosing a suitable gauge
condition, which is proved to be equivalent to a generalization of Adler's
condition, we apply the BRST gauge-fixing procedure to the Yang-Mills theory on
a hypersphere to obtain consistent results. Field equations for the Yang-Mills
field and associated fields are derived in manifestly O(n+1) covariant or
invariant forms. In the large radius limit, these equations reproduce the
corresponding field equations defined on the n-dimensional flat space.
