From the point of view of Cartan geometry, a Euclidean geometry $M$ is a Cartan geometry modeled over $(E(n),O(n))$ (the Euclidean group and orthogonal group respectively). The corresponding Klein geometry is called the Euclidean plane and since it is torsion-free (since it is flat) it is a Riemannian geometry.
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Author of the notes: Antonio J. Pan-Collantes
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