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By using the York canonical basis of ADM tetrad gravity, in a formulationusing radar 4-coordinates for the parametrization of the 3+1 splitting of thespace-time, it is possible to write the 4-Riemann tensor of a globallyhyperbolic, asymptotically Minkowskian space-time as a Hamiltonian tensor,whose components are 4-scalars with respect to the ordinary world4-coordinates, plus terms vanishing due to Einstein's equations. Therefore"on-shell" we find the expression of the Hamiltonian 4-Riemann tensor.Moreover, the 3+1 splitting of the space-time used to define the phase spaceallows us to introduce a Hamiltonian set of null tetrads and to find theHamiltonian expression of the 4-Ricci scalars of the Newman-Penrose formalism.  This material will be used in the second paper to study the 4-Weyl tensor,the 4-Weyl scalars and the four Weyl eigenvalues and to clarify the notions ofDirac and Bergmann observables.

Hamiltonian Expression of Curvature Tensors in the York Canonical Basis: I) The Riemann Tensor and Ricci Scalars