arXiv 1111.62602011-11-27math.DGmath.AP

关于黎曼叶理的一个共形积分不变量

A conformal integral invariant on Riemannian foliations

Guofang Wang, Yongbing Zhang

备注：: 10 pages
发表信息：: Proc. Amer. Math. Soc. 141 (2013), no. 4, 1405--1414

摘要：

设$M$是一个闭流形，它有一个余维为$q\geq 2$的叶状结构$\mathcal{F}$和一个类三角形度量$g_0$ .令$[g_0]_B$是类似于矩阵的度量的空间，其仅在水平方向上与$g_0$相差正基函数的倍数。假设$Y$是横截共形向量场，并且$(M,\mathcal{F},g_0)$的叶的平均曲率为零。我们表明，积分$\int_MY(R^T_{g^T})dμ_g$是独立的选择$g\in [g_0]_B$，其中$g^T$是横向度规引起的$g$和$R^T$是横向标量曲率。此外，如果$q\geq 3$，我们有任何$g\in [g_0]_B$的$\int_MY(R^T_{g^T})dμ_g=0$。但存在余维2极小黎曼叶理$(M,\mathcal{F},g)$和横截共形向量场$Y$使得$\int_MY(R^T_{g^T})dμ_g\neq 0$ .因此，它是一个非平凡的障碍横Yamabe问题上的最小黎曼叶理的余维2。

英文摘要：

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 $\int_MY(R^T_{g^T})dμ_g$ 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 $\int_MY(R^T_{g^T})dμ_g=0$ 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 $\int_MY(R^T_{g^T})dμ_g\neq 0$. Therefore, it is a nontrivial obstruction for the transverse Yamabe problem on minimal Riemannian foliation of codimension 2.

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