JB:"How can a geometry be Lorentzian?"
Huh?Come again? Lorentzian geometry (and not Riemannian geometry) is the basis of both special relativity and general relativity. How can you have such a geometry? Well, in lorentzian geometry you have a metric given by
(ds)^2= -(dt)^2+(dx)^2+(dy)^2+(dz)^2
instead of
(ds)^2=(dt)^2+(dx)^2+(dy)^2+(dz)^2
I am not exactly sure that I seee the problem.
JB:"Either you have a "flat space" with linearly independent coordinates or you have a bent space whatever shape you call it."
Well, the Riemaninan flat space (i.e. the Euclidean space) is given by a (global) metric g^ab=diag(+,+,+,+) while the Lorentzian flat space(i.e. the Minkovski space) is goven by a (global) metric g^ab=diag(-,+,+,+,). What is your point? I am not sure I follow.
JB:"Even Einstein's equations of general relativity describes a "bent space-time field" relative to a four-dimensional Euclidean space."
No, they originally have been developed for Lorentzian spacetimes. However, for certain applications where certain technicalities become annoying, you can have Euclidean/Riemanian GR.
JB:"In a Eucildean space of any dimension derivatives calculated relative to a change along one of the axes are always zero."
By no means. If what you say were true, you wouldn't have mechanics (classica) nor would you have differential geometry.
JB:"Thus within such a space one cannot calculate any change with time (the so-called fourth dimension); If nothing can change there must be nothing."
I would suggest that you take a look at lagrangean and hamiltonian mechanics. Also at quantum mechanics. Also differential equations and partial differential equations (specifically the initial value problem). I can give you refs on these issues if you so like.
Of course such a chnge in time can be calculated, that is why you have what is called the evolution equations (the equations of motion).
