Every open subset of Euclidean space R^{n} is an ndimensional Riemannian manifold in a natural manner: as charts for the manifold structure one can use identity maps, and the inner product structure comes from the dot product given on R^{n}.
If γ : [a, b] → M is a continuously differentiable curve in the Riemannian manifold M, then we define its length L(γ) by
With this definition of length, every connected Riemannian manifold M becomes a metric space in a natural fashion: the distance d(x, y) between the points x and y of M is defined as
Even though Riemannian manifolds can be "curved", there is still a notion of "straight line" on them: the geodesics. These are curves which locally join their points with shortest paths.
The Nash embedding theorem states that every Riemannian manifold M can be thought of as a submanifold of some Euclidean space R^{n}, with the notions of "length", "curvature" and "angle" on M coinciding with the ordinary ones in R^{n}.
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