Isometric Deformation of ( m , n )-Type Helicoidal Surface in the Three Dimensional Euclidean Space

We consider a new kind of helicoidal surface for natural numbers <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula> in the three-dimensional Euclidean space. We study a helicoidal surface of value <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>, which is locally isometric to a rotational surface of value <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mi>m</mi> <mo>,</mo> <mi>n</mi> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>. In addition, we calculate the Laplace&#8315;Beltrami operator of the rotational surface of value <inline-formula> <math display="inline"> <semantics> <mrow> <mo>(</mo> <mn>0</mn> <mo>,</mo> <mn>1</mn> <mo>)</mo> </mrow> </semantics> </math> </inline-formula>.

Abstract licence: CC BY

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