  Established fundamental physics can be described by fields, which are maps.
The source of such a map is space-time, which can be curved due to gravity. The
map itself needs to be curved in its gauge field part so as to describe
interaction forces like those mediated by photons and gluons. In the present
article, we permit non-zero curvature also on the internal space, the target of
the field map.
  The action functional and the symmetries are constructed in such a way that
they reduce to those of standard Yang-Mills-Higgs (YMH) gauge theories
precisely when the curvature on the target of the fields is turned off. For
curved targets one obtains a new theory, a curved YMH gauge theory. It realizes
in a mathematically consistent manner an old wish in the community: replacing
structures constants by functions depending on the scalars of the theory. In
addition, we provide a simple 4d toy model, where the gauge symmetry is
abelian, but turning off the gauge fields, no rigid symmetry remains---another
possible manifestation of target curvature.
  It now remains to be seen, if internal curvature in the above sense is
realized in nature. Curvature of space-time is proven, but still negligible in
particle physics, except for the very early universe where quantum gravity must
have played an essential role. An important question therefore is, if glimpses
of target curvature can be visible in accelerator physics. We know that at
contemporary energy scales, the usual (flat) standard model describes nature to
a very high accuracy. Could it be that the alleged deviations in the B to
D-star-tau-nu decay reported by BaBar in 2012 and recently also by LHCb are
already a manifestation of target curvature? What kind of effects does target
curvature have on a YMH theory in general, for what kind of effects do we need
to look out for so as to detect it?
