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I’ve been looking at the Wikipedia article onaffine connectionswith 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)’ (videGeorge 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?

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# Elementary questions about connections.

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