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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A conformal integral invariant on Riemannian foliations


Authors: Guofang Wang and Yongbing Zhang
Journal: Proc. Amer. Math. Soc. 141 (2013), 1405-1414
MSC (2010): Primary 53C21; Secondary 58E12, 53C12
Published electronically: September 7, 2012
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Abstract: Let $ M$ be a closed manifold which admits a foliation structure $ \mathcal {F}$ of codimension $ q\geq 2$ and a bundle-like metric $ g_0$. Let $ [g_0]_B$ be the space of bundle-like metrics which differ from $ g_0$ only along the horizontal directions by a multiple of a positive basic function. Assume $ Y$ is a transverse conformal vector field and the mean curvature of the leaves of $ (M,\mathcal {F},g_0)$ vanishes. We show that the integral $ \int _MY(R^T_{g^T})d\mu _g$ is independent of the choice of $ g\in [g_0]_B$, where $ g^T$ is the transverse metric induced by $ g$ and $ R^T$ is the transverse scalar curvature. Moreover if $ q\geq 3$, we have $ \int _MY(R^T_{g^T})d\mu _g=0$ for any $ g\in [g_0]_B$. However there exist codimension $ 2$ minimal Riemannian foliations $ (M,\mathcal {F},g)$ and transverse conformal vector fields $ Y$ such that $ \int _MY(R^T_{g^T})d\mu _g\neq 0$. Therefore, $ \int _MY(R^T_{g^T})d\mu _g$ is a nontrivial obstruction for the transverse Yamabe problem on minimal Riemannian foliation of codimension $ 2$.


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Additional Information

Guofang Wang
Affiliation: Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstr. 1, 79104 Freiburg, Germany
Email: guofang.wang@math.uni-freiburg.de

Yongbing Zhang
Affiliation: Department of Mathematics, University of Science and Technology of China, 230026 Hefei, People’s Republic of China – and – Mathematisches Institut, Albert-Ludwigs-Universität Freiburg, Eckerstr. 1, 79104 Freiburg, Germany
Email: ybzhang@amss.ac.cn

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11498-4
PII: S 0002-9939(2012)11498-4
Received by editor(s): August 21, 2011
Published electronically: September 7, 2012
Additional Notes: This project is supported by SFB/TR71 “Geometric partial differential equations” of DFG
Communicated by: Lei Ni
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.