Lichnerowicz and Obata theorems for foliations.
The authors prove that if \(M\) is a closed Riemannian manifold with a Riemannian foliation of codimension \(q\), and if the normal Ricci curvature satisfies \(\text{Ric}^\bot(X,X)\geq a(q-1)| X| ^2\) for every \(X\) in the normal bundle for some fixed \(a>0\), then the smallest eigenvalue \(\lambda_B\) of the basic Laplacian satisfies \(\lambda_B\geq aq\). In addition, if equality occurs, then the leaf space is isometric to the space of orbits of a discrete subgroup of \(O(q)\) acting on the standard \(q\)-sphere of constant sectional curvature \(a\). These results generalize the Lichnerowicz comparison theorem [\textit{A. Lichnerowicz}, Géométrie des groupes de transformations. (Travaux et recherches mathématiques 3. Paris: Dunod) (1958; Zbl 0096.16001)] and Obata's theorem [\textit{M. Obata}, J. Math. Soc. Japan 14, 333--340 (1962; Zbl 0115.39302)] for a closed Riemannian manifold \(M\) with the property \(\text{Ric}(X,X)\geq a(q-1)| X| ^2\) for every \(X\in TM\). A result about bundle-like metrics on foliations is also proved: On any Riemannian foliation with bundle-like metric, there exists a bundle-like metric for which the mean curvature is basic and the basic Laplacian for the new metric is the same as that of the original metric. Related to this paper is the research by \textit{J. J. Hebda} in [Indiana Univ. Math. J. 35, 321--331 (1986; Zbl 0596.53027)], where it is proved that \(\text{Ric}^\bot(X,X)\geq a(q-1)| X| ^2\) implies that the Riemannian foliation is taut (meaning that there exists a metric for which the leaves are minimal).
scientific article; zbMATH DE number 1868189