Invariants of Immersions on n-Dimensional Affine Manifold

Year 2024, Volume: 37 Issue: 2, 924 - 937, 01.06.2024

Djavvat Khadjiev , Gayrat Beshimov , İdris Ören

https://doi.org/10.35378/gujs.1037048

Abstract

Main results: The system of Christoffel symbols of the connection of an immersion ξ:J→R^n of an n-dimensional manifold J in the n-dimensional linear space R^n is a system of generators of the differential field of all Aff(n)-invariant differential rational functions of ξ, where Aff(n) is the group of all affine transformations of R^n. A similar result have obtained for the subgroup SAff(n) of ⁡Aff(n) generated by all unimodular linear transformations and parallel translations of R^n. Rigidity and uniqueness theorems for immersions ξ:J→R^n in geometries of groups Aff(n) and SAff(n) were obtained. These theorems are given in terms of the affine connection and the volume form of immersions.

Keywords

Connection, Riemannian curvature tensor, Invariant

Supporting Institution

The Scientific and Technological Research Council of Turkey The Ministry of Innovative Development of the Republic of Uzbekistan (MID Uzbekistan)

Thanks

This work is supported by The Scientific and Technological Research Council of Turkey (T\"{U}B{\.I}TAK) under Grant Number 119N613 and The Ministry of Innovative Development of the Republic of Uzbekistan (MID Uzbekistan) under Grant Number UT-OT-2020-2.

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