نشانندههاي ايزومتري موضعي‌ گروههاي لي‌ سه‌ بعدي

In this thesis, we study the problem of local isometric embeddings of three-dimensional Lie groups equipped with left-invariant Riemannian metrics. According to the Janet-Cartan theorem, any analytic n-dimensional Rieman-nian manifold can be locally isometrically embedded into a Euclidean space of dimension n(n+1) . However, determining the minimal dimension of the Euclidean 2space into which a given Riemannian manifold (M, g) can be embedded remains a challenging problem even when (M, g) is a homogeneous space. In this thesis, we investigate this problem for Riemannian manifolds (M, g) that are locally isometric to a three-dimensional Lie group with a left-invariant Riemannian metric. We classify all such spaces that can be embedded into four-dimensional Euclidean space. Our classification relies heavily on two key equations: the Gauss equation an‎d the derived Gauss equation.

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