  This is the first of a series of papers considering symmetry properties of
quantum systems over 2D graphs or manifolds, with continuous spins, in the
spirit of the Mermin--Wagner theorem. In the model considered here (quantum
rotators) the phase space of a single spin is a $d-$dimensional torus, and
spins (or particles) are attached to sites of a graph satisfying a special
bi-dimensionality property. The kinetic energy part of the Hamiltonian is minus
a half of the Laplace operator. We assume that the interaction potential is
C$^2$-smooth and invariant under the action of a connected Lie group ${\ttG}$.
A part of our approach is to give a definition (and a construction) of a class
of infinite-volume Gibbs states for the systems under consideration (the class
$\fG$). This class contains the so-called limit Gibbs states, with or without
boundary conditions. We use ideas and techniques originated from various past
papers, in combination with the Feynman--Kac representation, to prove that any
state lying in the class $\fG$ (defined in the text) is ${\ttG}$-invariant. An
example is given where the interaction potential is singular and there exists a
Gibbs state which is not ${\ttG}$-invariant.
  In the next paper under the same title we establish a similar result for a
bosonic model where particles can jump from a vertex of the graph to one of its
neighbors (a generalized Hubbard model).
