  Loop Quantum Gravity heavily relies on a connection formulation of General
Relativity such that 1. the connection Poisson commutes with itself and 2. the
corresponding gauge group is compact. This can be achieved starting from the
Palatini or Holst action when imposing the time gauge. Unfortunately, this
method is restricted to D+1 = 4 spacetime dimensions. However, interesting
String theories and Supergravity theories require higher dimensions and it
would therefore be desirable to have higher dimensional Supergravity loop
quantisations at one's disposal in order to compare these approaches. In this
series of papers, we take first steps towards this goal. The present first
paper develops a classical canonical platform for a higher dimensional
connection formulation of the purely gravitational sector. The new ingredient
is a different extension of the ADM phase space than the one used in LQG, which
does not require the time gauge and which generalises to any dimension D > 1.
The result is a Yang-Mills theory phase space subject to Gauss, spatial
diffeomorphism and Hamiltonian constraint as well as one additional constraint,
called the simplicity constraint. The structure group can be chosen to be
SO(1,D) or SO(D+1) and the latter choice is preferred for purposes of
quantisation.
