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Euclidean Geometry (Definition)

The classical geometry of flat space based on Euclid's axioms.
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The statement of the theorem

Let (M,g)(M, g) be a smooth, connected, nn-dimensional Riemannian manifold. The geometry defined by (M,g)(M, g) is Euclidean if and only if the sectional curvature KK of the manifold is identically zero everywhere. Formally, this means that for any point p∈Mp \in M and any two-dimensional subspace Π⊂TpM\Pi \subset T_p M (the tangent space at pp), the sectional curvature K(Π)K(\Pi) must satisfy:\n\nK(Π)=R(v,w,v,w)g(v,v)g(w,w)−g(v,w)2=0K(\Pi) = \frac{R(v, w, v, w)}{g(v, v)g(w, w) - g(v, w)^2} = 0\n\nwhere RR is the Riemann curvature tensor, and v,wv, w are tangent vectors spanning Π\Pi. Furthermore, the manifold (M,g)(M, g) must be locally isometric to the standard Euclidean space (Rn,gEuc)(\mathbb{R}^n, g_{Euc}), where gEucg_{Euc} is the standard Euclidean metric tensor δij\delta_{ij}. This condition ensures that the parallel postulate holds in the strongest sense.