Elementary questions about connections. I’ve been looking at the Wikipedia article on affine connections with a view to better understanding the choice of Riemannian geometry for our best theory of gravity — a founding choice made by Einstein nearly a century ago. Sadly for me the Wikipedia article seems to be have been written for ‘streamlined folk who think in slogans (here symbols) and talk with bullets (here formulae)’ (vide George Orwell). I’m un-streamlined and I’d therefore like to ask a few primitive questions unrestricted to the simple distortion (called curvature) of, say, a two-dimensional Euclidean space exemplified by a two-dimensional, uniformly curved spherical surface. Am I correct in supposing that for physics purposes, connections are used to extend algebraic descriptions of geometry and analyses needing calculus to beyond a ‘local’ region, inside which deviations from Euclidean geometry are imperceptible? If so, I guess that some other kinds of connection would be needed algebraically to describe and analyse, if deemed necessary, other distortions from Euclidean geometry. For example, distortions like that of a helical two-dimensional geometry, say a multi-storey car park floor-surface geometry. Or of the surfaces of Klein bottles and Moibus strips. Or of scale-changes, gauged by some agreed protocol, as in a lattice where temperature varies and expansion is measured with an Invar ruler or by counting lattice steps. In such cases, I suppose, not only could distortions classified as ‘curvature’ be measured by orientation changes of a vector carried around a closed circuit (by parallel transport using an affine connection) and specified by a Riemann tensor: but different distortions might produce different changes (like vertical discontinuities in a car park) in mathematical objects carried around circuits that would be expected to close in the ‘local’ Euclidean limit. Could someone tell me what the connections that would be used in such cases are called and what mathematical objects would be used to specify such distortions? And are connections other than 'affine and Levi-Civita' inappropriate for use in a theory of gravity because they would not yield a Newtonian inverse square law locally in a Euclidean limit? Or for a more streamlined reason?