  The relationship between minimal algebraic Kac-Moody groups and twin
buildings is well known as is the relationship between formal completions in
one direction and affine buildings. Nevertheless, as the completion of a
Kac-Moody group in one direction destroys the opposite BN-pair, there exists no
longer a twin building. For similar reasons, there are so far no buildings at
all for analytic completions of affine Kac-Moody groups. In this article we
construct a new type of twin buildings, called twin cities, that are associated
to affine analytic Kac-Moody groups over the real or complex numbers. Twin
cities consist of two sets of buildings. We describe applications of cities in
infinite dimensional differential geometry by proving infinite dimensional
versions of classical results from finite dimensional differential geometry:
For example, we show that points in an isoparametric submanifold in a Hilbert
space correspond to all chambers in a city. In a sequel we will describe the
theory of twin cities for formal completions.
