arXiv subjects
Publications and source records attributed to Hasan M. El-Hasan.
We prove a Tachibana-type result for Kähler-Einstein manifolds isolating complex projective space based on its Calabi curvature operator. The proof uses a new Bochner formula expressing the Lichnerowicz curvature term via the Calabi curvature operator. This resolves one of the problems posed at the AIM workshop "The Bochner Technique'' (May 2026).
Pro and the third author showed that there are Riemannian submersions $π: M \to B$ with $M$ a compact manifold with positive Ricci curvature, whose base $B$, has Ricci curvatures with both signs. Thus, Riemannian submersions need not preserve positive Ricci curvature. In this note we establish the degree to which this result extends into the setting of positive intermediate Ricci curvature. It is an immediate consequence of the Gray--O'Neill Horizontal curvature equation that if $π: M\to B$ is a Riemannian submersion whose base is $b$-dimensional and $\mathrm{Ric}_{k}(M) >0$ for any $k \in \{ 1,2,\cdots, b-1\}$, then $ \mathrm{Ric}_{k}(B)$ is also positive. Here we show that this observation is optimal in the following strong sense: For $k \geq \mathrm{dim}(B)$, let $π: (M,g_M) \to (B,g_B)$ be a Riemannian submersion from a complete Riemannian manifold with $\mathrm{Ric}_{k}(M) >0$. We show how to perturb $g_M$ in the $C^1$-topology to produce a Riemannian submersion $π: (M,\tilde{g}_M) \to (B,\tilde{g}_B)$ whose total space has $\mathrm{Ric}_{k} >0$, but whose base has Ricci curvature of both signs. In particular, this shows that Riemannian submersions that do not preserve positive Ricci curvature are dense in the $C^1$-topology among the complete metrics on $M$ with $\mathrm{Ric}>0$ for which a given submersion $π: M\to B$ is Riemannian.
Murphy and the second author showed that a generic closed Riemannian manifold has no totally geodesic submanifolds, provided it is at least four dimensional. Lytchak and Petrunin established the same thing in dimension 3. For the higher dimensional result, the generic set is open and dense in the $C^{q}$--topology for any $% q\geq 2.$ In Lytchak and Petrunin's work, the generic set is a dense $G_{δ}$ in the $C^{q}$-topology for any $q\geq 2.$ Here we show that the set of such metrics on a compact $3$-manifold contains a set that is open and dense in the $C^{q}$-topology for any $q\geq 3.$