LAUSR

Due to Janet–Cartan’s theorem, any analytic Riemannian manifolds can be locally isometrically embedded into a sufficiently high dimensional Euclidean space. However, for an individual Riemannian manifold ( M ,  g ), it is in general hard to determine the least dimensional Euclidean space into which ( M ,  g ) can be locally isometrically embedded, even in the case where ( M ,  g ) is homogeneous. In this paper, when the space ( M ,  g ) is locally isometric to a three-dimensional Lie group equipped with a left-invariant Riemannian metric, we classify all such spaces that can be locally isometrically embedded into the four-dimensional Euclidean space. Two types of algebraic equations, the Gauss equation and the derived Gauss equation, play an essential role in this classification.