  There are two mathematical relativity frameworks generalizing the black hole
theory: the theory of isolated horizons (IH) and the theory of near horizon
geometries (NHG). We outline here and discuss the derivation of the NHG from
the theory of IH by composing spacetimes from IH. The simplest but still quite
general class of solutions to Einstein's equations of this type defines
spacetimes foliated by Killing horizons emanating from extremal horizons. That
derivation, clearly being a link between the two frameworks, seems to be
unknown to the NHG researchers and is hardly acknowledged in reviews on the IH.
This lecture was a contribution to the Mathematical Structures session of the
2nd LeCosPA International Symposium "Everything about Gravity" celebrating the
centenary of Einstein's General Relativity on December 14-18, 2015 in Taipei.
