\(T\)-structure and the Yamabe invariant
type: Item
instance of: scholarly article
zbMATH Open document ID: 1239.53052
DOI: 10.4134/BKMS.2012.49.2.435
author: Chanyoung Sung
published in: Bulletin of the Korean Mathematical Society
publication date: 2012-04-13T00:00:00Z
full work available at URL: https://arxiv.org/abs/0812.4508
full work available at URL: http://www.mathnet.or.kr/mathnet/kms_content.php
Mathematics Subject Classification ID: 53C20
Mathematics Subject Classification ID: 55R10
zbMATH DE Number: 6023973
zbMATH Keywords: Yamabe invariant
zbMATH Keywords: \(T\)-structure
zbMATH Keywords: torus bundle
MaRDI profile type: Publication
arXiv ID: 0812.4508
OpenAlex ID: W2964135664
title: \(T\)-structure and the Yamabe invariant
review text: The Yamabe invariant, introduced by the author [Differ. Geom. Appl. 24, No. 3, 271--287 (2006; Zbl 1099.53031); J. Geom. Phys. 59, No. 2, 246--255 (2009; Zbl 1163.53023)], is an invariant of a smooth closed manifold depending on its smooth topology. Let \(M\) be a smooth closed manifold of dimension \(n\). Given a smooth Riemannian metric \(g\) on it, the conformal class \([g]\) is defined as NEWLINE\([g] = \{\varphi g \mid \varphi:M \rightarrow \mathbb{R}^{+}text{ is smooth}\}\). The famous Yamabe problem [\textit{J. Lee} and \textit{T. Parker}, ``The Yamabe problem'', Bull. Am. Math. Soc., New Ser. 17, 37--91 (1987; Zbl 0633.53062)] states that there exists a metric \(\widetilde{g}\) in \([g]\) which attains the minimum NEWLINE\[NEWLINE \inf _{\widetilde{g}\in [g]}\frac{\int_M s_{\widetilde{g}}dV_{\widetilde{g}}}{(\int_M dV_{\widetilde{g}})^ {\frac{n-2}{n}}},NEWLINE\]NEWLINE where \(s_{\widetilde{g}}\) and \(dV_{\widetilde{g}}\) denote the scalar curvature and the volume element of \(\widetilde{g}\). NEWLINENEWLINENEWLINE NEWLINEIn turns out that when \(n\geq 3\), a unit-volume minimizer \(\widetilde{g}\) in \([g]\) has constant scalar curvature, which is equal to the above minimum value called the Yamabe constant of \([g]\) and denoted by \(Y(M,[g])\). It is known that the Yamabe constant of any \(n\)-manifold is bounded above by \(Y(S^n,[g_0])\) where \([g_0]\) denotes a standard round metric. Thus following a min-max procedure the author defines the Yamabe invariant NEWLINE\[NEWLINEY(M):=\sup_{[g]}Y(M,[g])NEWLINE\]NEWLINE on \(M\). The Yamabe invariant contains information about possible scalar curvature on it. For example:NEWLINENEWLINE\(\bullet\) \(Y(M)>0\) if and only if \(M\) admits a metric of positive scalar curvature.NEWLINENEWLINE\(\bullet\) If \(M\) is simply-connected and dim \(M\geq 5\), then \(Y(M)\geq 0\). With the further assumption that \(M\) is spin, \(Y>0\) if and only if the \(\alpha\)-genus of \(M\) is 0, etc. In particular, it is well-known …
reviewed by: Ioan Pop
description: scientific article; zbMATH DE number 6023973