  The sine-Gordon equation has hyperelliptic al function solutions over a
hyperelliptic Jacobian for $y^2 = f(x)$ of arbitrary genus $g$. This article
gives an extension of the sine-Gordon equation to that over subvarieties of the
hyperelliptic Jacobian.
  We also obtain the condition that the sine-Gordon equation in a proper
subvariety of the Jacobian is defined.
