  Some hyperbolic systems are known to include implicit preservation of
differential constraints: these are for example the time conservation of the
curl or the divergence of a vector that appear as an implicit constraint. In
this article, we show that this kind of constraint can be easily conserved at
the discrete level with the classical discontinuous Galerkin method, provided
the right approximation space is used for the vectorial space, and under some
mild assumption on the numerical flux. For this, we develop a discrete
differential geometry framework for some well chosen piece-wise polynomial
vector approximation space. More precisely, we define the discrete Hodge star
operator, the exterior derivative, and their adjoints. The discrete adjoint
divergence and curl are proven to be exactly preserved by the discontinuous
Galerkin method under a small assumption on the numerical flux. Numerical tests
are performed on the wave system, the two dimensional Maxwell system and the
induction equation, and confirm that the differential constraints are preserved
at machine precision while keeping the high order of accuracy.
