  Based on its central role in the framework of real calculi, the existence of
the Levi-Civita connection for real calculi over projective modules is studied,
with a special emphasis placed on the simple module of N-dimensional complex
vectors over the algebra of complex N-by-N matrices. It is demonstrated that
existence of the Levi-Civita connection in this case depends on the Lie algebra
g of hermitian derivations, and necessary and sufficient conditions for the
possibility of constructing a real calculus for which there exists a
Levi-Civita connection are given in this case. Furthermore, in the general case
of real calculi over projective modules, necessary and sufficient conditions
for the existence of the Levi-Civita connection are given in terms of explicit
projection coefficients.
