Clairaut Submersion

In this chapter, we give the detailed study about the Clairaut submersion. The fundamental notations are given. Clairaut submersion is one of the most interesting topics in differential geometry. Depending on the condition on distribution of submersion, we have different classes of submersion such as anti-invariant, semi-invariant submersions etc. We describe the geometric properties of Clairaut anti-invariant submersions and Clairaut semi-invariant submersions whose total space is a Kähler, nearly Kähler manifold. We give condition for Clairaut anti-invariant submersion to be a totally geodesic map and also study Clairaut anti-invariant submersions with totally umbilical fibers. We also give the conditions for the semi-invariant submersions to be Clairaut map and also for Clairaut semi-invariant submersion to be a totally geodesic map. We also give some illustrative example of Clairaut anti-invariant and semi-invariant submersion.

Topological volumes of fibrations: a note on open covers

We establish a straightforward estimate for the number of open sets with fundamental group constraints needed to cover the total space of fibrations. This leads to vanishing results for simplicial volume and minimal volume entropy, e.g., for certain mapping tori.

A β-tensor on Kaehler manifolds and its geometric characterizations

In this paper, we first introduce a new notion [Formula: see text]-tensor on Hermitian manifold and particularly, we present some geometric characterizations of such a tensor on the Kaehler manifold. Here, we investigate the Kaehler submersion whose total space is equipped with the [Formula: see text]-tensor and obtain some results. Also, we deal with a Kaehler submersion with totally geodesic fibers such that the total space admits [Formula: see text]-Ricci soliton and [Formula: see text]-tensor. Finally, we give necessary conditions for which any fiber and base manifold of Kaehler submersion is [Formula: see text]-Ricci soliton or [Formula: see text]-Kaehler-Einstein.

Complex hyperkähler structures defined by Donaldson–Thomas invariants

The notion of a Joyce structure was introduced in Bridgeland (Geometry from Donaldson–Thomas invariants, preprint arXiv:1912.06504) to describe the geometric structure on the space of stability conditions of a $$\hbox {CY}_3$$ CY 3 category naturally encoded by the Donaldson-Thomas invariants. In this paper we show that a Joyce structure on a complex manifold defines a complex hyperkähler structure on the total space of its tangent bundle, and give a characterisation of the resulting hyperkähler metrics in geometric terms.

A support theorem for Hilbert schemes of planar curves, II

We study the cohomology of Jacobians and Hilbert schemes of points on reduced and locally planar curves, which are however allowed to be singular and reducible. We show that the cohomologies of all Hilbert schemes of all subcurves are encoded in the cohomologies of the fine compactified Jacobians of connected subcurves, via the perverse Leray filtration. We also prove, along the way, a result of independent interest, giving sufficient conditions for smoothness of the total space of the relative compactified Jacobian of a family of locally planar curves.

Rigidity of the mod 2 families Seiberg–Witten invariants and topology of families of spin 4-manifolds

We show a rigidity theorem for the Seiberg–Witten invariants mod 2 for families of spin 4-manifolds. A mechanism of this rigidity theorem also gives a family version of 10/8-type inequality. As an application, we prove the existence of non-smoothable topological families of 4-manifolds whose fiber, base space, and total space are smoothable as manifolds. These non-smoothable topological families provide new examples of $4$ -manifolds $M$ for which the inclusion maps $\operatorname {Diff}(M) \hookrightarrow \operatorname {Homeo}(M)$ are not weak homotopy equivalences. We shall also give a new series of non-smoothable topological actions on some spin $4$ -manifolds.

Spectral Estimates for Riemannian Submersions with Fibers of Basic Mean Curvature

For Riemannian submersions with fibers of basic mean curvature, we compare the spectrum of the total space with the spectrum of a Schrödinger operator on the base manifold. Exploiting this concept, we study submersions arising from actions of Lie groups. In this context, we extend the state-of-the-art results on the bottom of the spectrum under Riemannian coverings. As an application, we compute the bottom of the spectrum and the Cheeger constant of connected, amenable Lie groups.