  A stationary observer in the Kerr spacetime has zero angular momentum if
his/her angular velocity $\omega$ has a particular value, which depends on the
position of the observer. Worldlines of such zero angular momentum observers
(ZAMOs) with the same value of the angular velocity $\omega$ form a three
dimensional surface, which has the property that the Killing vectors generating
time translation and rotation are tangent to it. We call such a surface a
rigidly rotating ZAMO surface. This definition allows a natural generalization
to the surfaces inside the black hole, where ZAMO's trajectories formally
become spacelike. A general property of such a surface is that there exist
linear combinations of the Killing vectors with constant coefficients which
make them orthogonal on it. In this paper we discuss properties of the rigidly
rotating ZAMO surfaces both outside and inside the black hole and relevance of
these objects to a couple of interesting physical problems.
