I have just began my journey in General Theory of relativity through the book of Weinberg-Gravitation and cosmology principles and applications of the general theory of relativity I don't know if this topic is just a repeat of an older one, if it is please let me know the link to it... I started reading about the Principle of Equivalence, and it somehow confused me. It says that the Strong P.o.E tells us that: At every spacetime point in an arbitary gravitational field it is possible to choose a "locally inertial coordinate sysyem" such that within sufficiently small region of the point in question, laws of nature take the same form they have in unaccelerated Cartesian Coordinate systems in the absence of gravity (meaning the laws we get from special relativity). At this point I thought like this: If for example I get a spacetime point x in an arbitrary curved space of metric gαβ, then the metric on that point is equal to the metric of a Minkowski space (Gauss's curvature is equal to 0- flat space). This would mean that: gαβ(x)=nαβ(x) Going to another point on the same space, point X'=x+Δx, I can still use the P.o.E. to write on that point: gαβ(X')=nαβ(X') Doesn't that imply that ,by keeping doing the same work over and over again, a space of an arbitrary curvature can in fact be described by a flat space? That is wrong (for example the Sphere cannot). Where is the "gap" in my approach and so my understanding?