  The aim of this paper is to give a new method to construct explicit formulas
for algebraic differential operators of any order on a finitely generated
projective module $E$ on a commutative unital ring $A$. We moreover give
explicit formulas for algebraic connections on a class of finitely generated
projective modules on ellipsoid surfaces. The connections we construct are
non-flat with trace of curvature equal to zero. We construct these formulas
using an idempotent matrix $M$ defining the module $E$. Such an idempotent
matrix $M$ is constructed from a "projective basis" $B$ defining the module
$E$. Associated to a projective basis $B$ for $E$ we construct a connection
$\nabla_B$. The curvature $R_{\nabla_B}$ of the connection $\nabla_B$ is given
by a Lie product: $R_{\nabla_B}(x,y):=[\nabla_B(x)(M), \nabla_B(y)(M)]$
involving the matrix $M$, and this Lie product is non-zero in general. Hence
the curvature formula indicates that most projective finite rank modules do not
have a flat algebraic connection. We also give an explicit formula for a
non-flat algebraic connection on the cotangent bundle $\Omega$ of the real
2-sphere. The cotangent bundle $\Omega$ is topologically non-trivial and it is
not clear if it has a flat algebraic connection. All higher Chern classes in
deRham cohomology are zero: $c_i(\Omega)=0$ for all $i \geq 1$. We relate the
construction to non-abelian extensions and a refined characteristic class
$c(\Omega)$ introduced in another paper on the subject. The class $c(\Omega)$
is defined using the connection $\nabla_B$ but it is independent of choice of
connection. The class $c(-)$ lives in a torsor. The methods introduced in the
paper prove that the underlying complex manifold of any complex affine regular
hypersurface is a Calabi-Yau manifold. This is because its canonical bundle is
trivial.
