Geometry of extended Bianchi-Cartan-Vranceanu spaces

TL;DR

This paper introduces extended Bianchi-Cartan-Vranceanu spaces as a seven-dimensional generalization of classical 3D spaces, analyzing their geometric properties like connections, curvatures, symmetries, and geodesics.

Contribution

It presents the first study of extended Bianchi-Cartan-Vranceanu spaces, expanding the classical geometry to higher dimensions and exploring their fundamental geometric features.

Findings

Derived the Levi-Civita connection for EBCV spaces

Computed Ricci curvatures of EBCV spaces

Identified Killing fields and analyzed geodesics in EBCV spaces

Abstract

The differential geometry of $3$-dimensional Bianchi, Cartan and Vranceanu ($BCV$) spaces is well known. We introduce the extended Bianchi, Cartan and Vranceanu ($EBCV$) spaces as a natural seven dimensional generalization of $BCV$ spaces and study some of their main geometric properties, such as the Levi-Civita connection, Ricci curvatures, Killing fields and geodesics.