  We present a coordinate-free background space construction of Euclidean
Jackiw-Teitelboim gravity. It is written as a gauge theory that utilizes the
Killing vectors and conformal Killing vectors of a hyperboloid embedded in a
three dimensional background. A novel feature of the gauge theory is that
vanishing field strength does not necessarily imply that the gauge potentials
are pure gauges, not even locally. As is usual, metric tensors are dynamically
generated from the classical solutions of the theory, which here do not rely on
coordinate charts on the two-dimensional surface. We find a special class of
solutions whereby the derived metric tensor on the surface is the induced
metric from the background space. The gauge theory construction given here has
a natural generalization to a non-commutative space, which does not require the
use of coordinates, symbols or a star product.
