Abstract
We find all the flat surfaces in the unit 3-sphere \(\mathbb{S}^{3}\) that pass through a given regular curve of \(\mathbb{S}^{3}\) with a prescribed tangent plane distribution along this curve. The formula that solves this problem may be seen as a geometric analogue of the classical D’Alembert formula that solves the Cauchy problem for the homogeneous wave equation. We also provide several applications of this geometric D’Alembert formula, including a classification of the flat Mobius strips of \(\mathbb{S}^{3}\) .
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