  We formulate conditions on the geometry of a non-expanding horizon $\Delta$
which are sufficient for the space-time metric to coincide on $\Delta$ with the
Kerr metric. We introduce an invariant which can be used as a measure of how
different the geometry of a given non-expanding horizon is from the geometry of
the Kerr horizon. Directly, our results concern the space-time metric at $\IH$
at the zeroth and the first orders. Combained with the results of Ashtekar,
Beetle and Lewandowski, our conditions can be used to compare the space-time
geometry at the non-expanding horizon with that of Kerr to every order. The
results should be useful to numerical relativity in analyzing the sense in
which the final black hole horizon produced by a collapse or a merger
approaches the Kerr horizon.
