An immersed surface in $${{\mathbb {R}}}^4$$ is said to has constant Jordan angles (CJA) if the angles between its tangent planes and a fixed plane do not depend on the choice of the point. The constant Jordan angles surfaces in $${{\mathbb {R}}}^4$$ has been proved to exist, Bayard et al. (Geom Dedicata 162:153–176, 2013), but there are only explicit examples of non planar surfaces for the extremal angles 0 and $$\frac{\pi }{2}$$ as the Clifford torus. In this work, an alternative proof of the existence has been obtained that is based on the solution of a hyperbolic partial differential equation. Finally, after a study of the known solutions of the hyperbolic equation, an explicit expression with arbitrary CJA is provided for a family of immersions with an additional geometric property written in terms of the local invariants.

中文翻译：

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$${{\mathbb {R}}}^4$$R4 中表面的显式浸入，任意常数 Jordan 角

$${{\mathbb {R}}}^4$$ 中的浸没表面如果其切平面和固定平面之间的角度不取决于点的选择，则称其具有恒定的乔丹角（CJA）。Bayard 等人已证明 $${{\mathbb {R}}}^4$$ 中的恒定乔丹角曲面是存在的。(Geom Dedicata 162:153–176, 2013)，但只有极角 0 和 $$\frac{\pi }{2}$$ 作为 Clifford 环面的非平面表面的明确示例。在这项工作中，已经获得了基于双曲偏微分方程解的另一种存在证明。最后，在研究双曲方程的已知解之后，为具有附加几何属性的浸入族提供了具有任意 CJA 的显式表达式，该表达式以局部不变量形式表示。