Geodesic
Weingarten equations for surfaces on Helmholtz-type groups
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXV", Voronezh, April 26-30, 2024, Part 1, Tome 235 (2024), pp. 68-77

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In this paper, we study surfaces on three-dimensional Helmholtz-type Lie groups that define the actions of groups of motions of Helmholtz geometries, which are geometries of local maximal mobility. In this paper, we present left-invariant metrics and Levi-Civita connections for these Lie groups, which were found earlier. For surfaces of Helmholtz-type Lie groups, we calculate the spinors that generate them, which satisfy the Dirac and Weingarten equations. We also derive compatibility conditions for the Weingarten equations.
Mots-clés : Lie group, surface on a Lie group, Dirac equation, Weingarten equation
Keywords: Codazzi equation 
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     title = {Weingarten equations for surfaces on {Helmholtz-type} groups},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {68--77},
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     volume = {235},
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V. A. Kyrov. Weingarten equations for surfaces on Helmholtz-type groups. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Proceedings of the Voronezh international spring mathematical school "Modern methods of the theory of boundary-value problems. Pontryagin readings—XXXV", Voronezh, April 26-30, 2024, Part 1, Tome 235 (2024), pp. 68-77. http://geodesic.mathdoc.fr/item/INTO_2024_235_a5/