On the first eigenvalue of non-orientable closed surfaces
type: Item
instance of: scholarly article
title: On the first eigenvalue of non-orientable closed surfaces
zbMATH Open document ID: 0932.58035
author: Katsuhiro Yoshiji
published in: Tsukuba Journal of Mathematics
publication date: 2000-03-06T00:00:00Z
review text: Let \( (M, g) \) be a 2-dimensional non-orientable closed Riemannian manifold and \( (\widetilde{M}, \widetilde{g}) \) its orientable Riemannian double cover. Let \(\lambda_1(M,g)\) (resp. \(\lambda_1(\widetilde{M},\widetilde{g})\)) denote the first nonzero eigenvalues of the Laplacian for functions on \((M,g)\) (resp. on \((\widetilde{M},\widetilde{g})\)). The author proves the following two results: (i) If \( M \) is homeomorphic to \( \mathbb{R} P ^2 , \) then \( \lambda _1 (M, g) > \lambda _1 ( \widetilde{M}, \widetilde{g}) \) for every metric \( g \) on \( M .\) (ii) If \( M \) is homeomorphic to \( \# ^n \mathbb{R} P ^2\) (the connected sum of \(n\) copies of the real projective plane) \((n \geq 2)\), there exists a metric \( g \) on \( M \) such that \( \lambda _1 (M, g) = \lambda _1 ( \widetilde{M}, \widetilde{g})\).
Mathematics Subject Classification ID: 58J50
zbMATH DE Number: 1309330
zbMATH Keywords: first eigenvalue
zbMATH Keywords: orientable Riemannian double cover
zbMATH Keywords: Riemannian manifold
reviewed by: Mircea Puta
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.21099/tkbjm/1496163676
OpenAlex ID: W640766416
DOI: 10.21099/TKBJM/1496163676
description: scientific article; zbMATH DE number 1309330