... three-dimensional manifold, an open set Ω ⊂ R3 equipped with an immersion Θ : Ω → E3 becomes an example of a Riemannian manifold (Ω; (gij )), i.e., a manifold, the set Ω, equipped with a Riemannian metric, ... generally, a Riemannian metric on a manifold is a twice covariant, symmetric, positive-definite tensor field acting on vectors in the tangent spaces to the manifold (these tangent spaces coincide with R3 ... open subset of R3 Then a Riemannian manifold (Ω; (gij )) with a Riemannian metric (gij ) of class C in Ω is flat if and only if its Riemannian curvature tensor vanishes in Ω Recast as such, this...