On local isometric embeddings of three-dimensional Lie groups

TL;DR

This paper classifies three-dimensional Lie groups with left-invariant metrics that can be locally embedded into four-dimensional Euclidean space, using algebraic equations related to the Gauss equation.

Contribution

It provides a complete classification of such Lie groups, identifying which can be locally embedded into -dimensional Euclidean space.

Findings

Classified all 3D Lie groups locally embeddable into D space.

Identified algebraic conditions involving Gauss equations for embeddings.

Enhanced understanding of geometric structures of Lie groups with invariant metrics.

Abstract

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.