Infinitesimal rotary transformation

Abstract

The paper is devoted to further study of a certain type of infinitesimal transformations of twodimensional (pseudo-) Riemannian spaces, which are called rotary. An infinitesimal transformation is called rotary if it maps any geodesic on (pseudo-) Riemannian space onto an isoperimetric extremal of rotation in their principal parts on (pseudo-) Riemannian space. We study basic equations of the infinitesimal rotary transformations in detail and obtain the simpler fundamental equations of these transformations.

The total torsion of knots under second order infinitesimal bending

“…Infinitesimal bending of curves and surfaces is studied, for instance, in [2,6,12,13,14]. Some results on infinitesimal transformations can be found in [3,5,11]. Infinitesimal bending of knots is considered in [4,7,10].…”

Infinitesimal bending of DNA helices

“…During the infinitesimal bending of curves are obtained the ruled surfaces [5]. More about infinitesimal deformation theory can be found in [1,4,7,8,10,11,13,14].…”

Curves on ruled surfaces under infinitesimal bending

“…Many other geometric magnitudes stay invariant in the sense that they don't get the variations of the first order (for example, the coefficients of the first fundamental form, Cristoffel's symbols, Gaussian curvature etc.). Many papers are devoted to the infinitesimal bending of curves, surfaces and manifolds (see (Aleksandrov, 1936;Efimov, 1948;Kon-Fossen, 1959;Vekua, 1959;Ivanova-Karatopraklieva & Sabitov, 1995;Velimirović, 2001a,b;Hinterleitner et al, 2008;Rančić et al, 2009;Alexandrov, 2010;Najdanović, 2015;Najdanović & Velimirović, 2017;Kauffman et al, 2019;Najdanović et al, 2019;Rančić et al, 2019;Rýparová & Mikeš, 2019;Belova et al, 2021;Maksimović et al, 2021).…”