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added the following to the References-section of the entries Euclidean geometry, synthetic geometry and Coq:
A textbook account of the axiomatization of Euclidean geometry is
- Wolfram Schwabhäuser, W.Szmielew, Alfred Tarski, Mathematische Methoden in der Geometrie, Springer 1983
Full formalization of this book in Coq (as synthetic geometry but following Tarski’s work) is discussed at
I think that the idea sentence including
Euclidean geometry can be regarded as the local model for Riemannian geometry, in some sense
is plainly wrong and misleading. Unlike the symplectic manifolds where the Darboux theorem has all such reducing to a single model locally, the Riemann geometries are locally nonisomorphic, hence Euclidean geometry can not be considered as THE local model, but just one of the very special flat cases. On the other side, Euclidean spaces have many global symmetries which play a major role in that geometry, which do not hold in Riemannian geometries which are equivalent to it locally, what also makes the statement very misleading.
When one says, as is standard, that Klein geometries are local models for Cartan geometries, this refers to tangent spaces. The further identification over open neighbourhoods (that is possible for symplectic geometry) is an extra integrability condition.
Maybe you would like to say “infinitesimal local model”, and feel invited to add this to the entry, but I think just “local model” here is standard terminology.
I vote for infinitesimal local model.
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