Please use this identifier to cite or link to this item:
https://hdl.handle.net/10316/112197
Title: | Curvature-adapted submanifolds of semi-Riemannian groups | Authors: | Camarinha, Margarida Raffaelli, Matteo |
Keywords: | Mathematics - Differential Geometry; Mathematics - Differential Geometry; 53C40 (Primary) 53B25, 53C30 (Secondary) | Issue Date: | 27-Mar-2020 | Publisher: | World Scientific | Serial title, monograph or event: | International Journal of Mathematics | Volume: | 34 | Issue: | 09 | Abstract: | We study semi-Riemannian submanifolds of arbitrary codimension in a Lie group $G$ equipped with a bi-invariant metric. In particular, we show that, if the normal bundle of $M \subset G$ is closed under the Lie bracket, then any normal Jacobi operator $K$ of $M$ equals the square of the associated invariant shape operator $\alpha$. This permits to understand curvature adaptedness to $G$ geometrically, in terms of left translations. For example, in the case where $M$ is a Riemannian hypersurface, our main result states that the normal Jacobi operator commutes with the ordinary shape operator precisely when the left-invariant extension of each of its eigenspaces has first-order tangency with $M$ along all the others. As a further consequence of the equality $K = \alpha^{2}$, we obtain a new case-independent proof of a well-known fact: every three-dimensional Lie group equipped with a bi-invariant semi-Riemannian metric has constant curvature. | Description: | 12 pages, no figures. Some changes in section 1; Theorem 1.5 and Corollary 1.7 corrected. To appear in International Journal of Mathematics | URI: | https://hdl.handle.net/10316/112197 | ISSN: | 0129-167X 1793-6519 |
DOI: | 10.1142/S0129167X23500532 | Rights: | openAccess |
Appears in Collections: | FCTUC Matemática - Artigos em Revistas Internacionais I&D CMUC - Artigos em Revistas Internacionais |
Files in This Item:
File | Description | Size | Format | |
---|---|---|---|---|
Curvature-adapted submanifolds of semi-Riemannian groups.pdf | 379.7 kB | Adobe PDF | View/Open |
Page view(s)
37
checked on Sep 11, 2024
Download(s)
33
checked on Sep 11, 2024
Google ScholarTM
Check
Altmetric
Altmetric
This item is licensed under a Creative Commons License