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A simple interpretation of the PDE for isometric immersion of the hyperbolic plane  ℍ 2 into  ℝ 4 is given. Thus, necessary and sufficient conditions are obtained. And, under natural additional conditions, we show that there are no complete solutions, but we can have special local solutions.

journal of mathematics

A PDE for Smooth Isometric Immersion of an Open Set of the Hyperbolic Plane <math xmlns="http://www.w3.org/1998/Math/MathML" id="M1"> <msup> <mrow> <mi mathvariant="normal">ℍ</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> </math> into <math xmlns="http://www.w3.org/1998/Math/MathML" id="M2"> <msup> <mrow> <mi>ℝ</mi> </mrow> <mrow> <mn>4</mn> </mrow> </msup> </math>

A PDE for Smooth Isometric Immersion of an Open Set of the Hyperbolic Plane ℍ 2 into ℝ 4

A PDE for Smooth Isometric Immersion of an Open Set of the Hyperbolic Plane $ℍ^{2}$ into $ℝ^{4}$