In this talk, we study isometric immersions of surfaces into simply connected three-dimensional unimodular Lie groups endowed with left-invariant metrics, both in the Riemannian and Lorentzian settings (assuming in the latter case that Milnor’s operator is diagonalizable). We present global coordinate models for these metric Lie groups, depending analytically on their structure constants, and discuss several fundamental results that characterize such immersions. In particular, we investigate to what extent an immersion can be reconstructed from natural geometric data, including (a) the tangent projections of a left-invariant frame, (b) the left-invariant Gauss map, and (c) the shape operator. As a main result, we show that an isometric immersion is essentially determined by its left-invariant Gauss map, up to certain well-controlled angular companions, which are characterized in terms of a PDE.

This is a joint work with I. Castro and J. S. Santiago available at https://arxiv.org/abs/2507.16728.

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