On Riemannian foliations over positively curved manifolds (Q1710254)

The goal of the paper under review is to present an obstruction to the existence of a metric $g$ of positive sectional curvature on a compact Riemannian manifold $M$ under the assumption that $(M, g)$ admits a foliation $\mathcal{F}$ with locally equidistant leaves (a Riemannian foliation). \par Let $(M, g)$ be a Riemannian manifold and $\mathcal{F}$ be a Riemannian foliation on $M$. The fundamental tensors of $\mathcal{F}$ are $A_X Y = \frac{1}{2}[X, Y]^v$ and $S_X \xi = -\nabla^v_\xi X$, where $X$ and $Y$ are \textit{horizontal} (orthogonal to $\mathcal{F}$) vector fields, and $\xi$ is a \textit{vertical} (tangent to $\mathcal{F}$) vector field. Let $c$ be a horizontal curve in $M$, then a vertical vector field $\xi$ is called a \textit{holonomy field} if $\nabla_{\dot{c}}\xi = -A^*_{\dot{c}}\xi - S_{\dot{c}}\xi$. A Riemannian foliation has bounded holonomy if there is a constant $L$ such that for every holonomy field $\xi$ and every $t$, $\| \xi(t) \| \le L \| \xi(0) \|$ (for the geometry of Riemannian foliations see [\textit{D. Gromoll} and \textit{G. Walschap}, Metric foliations and curvature. Basel: Birkhäuser (2009; Zbl 1163.53001)]). \par The main result of the paper is the following theorem: Let $\mathcal{F}$ be an odd-codimensional Riemannian foliation with bounded holonomy on a compact Riemannian manifold $(M, g)$. Then, at some point $p \in M$ there exists a plane $L \subset T_p M$ spanned by a unit vertical vector and a unit horizontal vector such that the sectional curvature $K(L)$ is non-positive. \par The author gives various applications of this theorem. For example, from the theorem one can obtain the classical result of Berger that every Killing field of a compact even-dimensional Riemannian manifold with positive sectional curvature has a zero. The theorem implies that Wilhelm's conjecture (If $\pi : (M, g_M) \to (B, g_B)$ is a Riemannian submersion, $\dim M = n + k$, $\dim B = n$, and $(M, g_M)$ is compact and has positive sectional curvature, then $k < n$) holds in case the Riemannian submersion has compact holonomy group, and $n$ is odd. As a corollary of the theorem the author obtains that if a Riemannian foliation $\mathcal{F}$ on a compact Riemannian manifold $(M, g)$ is principal (the leaves of $\mathcal{F}$ are the orbits of an isometric action of a Lie group $G$ with discrete isotropy subgroups), then the rank of $G$ is equal to $1$. \par As another application, the author considers Riemannian foliations $\mathcal{F}$ with totally geodesic leaves on $(M, g_0)$. For any basic function $\phi$, one can define the metric $g_\phi(X + \xi, X + \xi) = g_0(X, X) + e^{2\phi} g_0(\xi, \xi)$, where $X$ is horizontal and $\xi$ is vertical. The author proves that ``if $g_\phi$ has positive sectional curvature for some basic function $\phi$, then at some point $p \in M$, $A^*_X\xi$ is non-zero for all non-zero horizontal $X$ and vertical $\xi$ ($p$ is a `fat' point).'' This implies that in this case ``the codimension of $\mathcal{F}$ is even, and the dimension of $M$ is smaller than twice the codimension of $\mathcal{F}$.''