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We describe the isometry group of $L^2(\Omega, M)$ for Riemannian manifolds $M$ of dimension at least two with irreducible universal cover. We establish a rigidity result for the isometries of these spaces: any isometry arises from an…

For $g\geq 2$, let $\Gamma\subset\mathrm{Sp}(2g,\mathbb{R})$ be a discrete subgroup, which is either a cocompact subgroup or an arithmetic subgroup without torsion elements, and let $\mathbb{H}_{g}$ denote the Siegel upper half space of…

We show that the isometries of the manifold of scalars in $N=2$ supergravity in $d=5$ space-time dimensions can be broken by the supergravity interactions. The opposite conclusion holds for the dimensionally reduced $d=4$ theories, where…

The universal Spencer and de Rham complexes of sheaves over a smooth or analytical manifold are well known to play a basic role in the theory of $\mathcal{D}$-modules. In this article we consider a double complex of sheaves generalizing…

Let $E\to B$ be a complex analytic fiber bundle with fiber $F$, a flag variety over a compact complex manifold $B$. We shall obtain a description of the cohomology of $E$ when $B=X_\Gamma:=\Gamma\backslash X, E=Y_\Gamma:=\Gamma\backslash Y$…

Semi-Riemannian manifolds that satisfy (homogeneous) linear differential conditions of arbitrary order on the curvature are analyzed. They include, in particular, the spaces with (higher-order) recurrent curvature, (higher-order) symmetric…

We extend our previous analysis of Riemannian four-manifolds M admitting rigid supersymmetry to N=1 theories that do not possess a U(1)_R symmetry. With one exception, we find that M must be a Hermitian manifold. However, the presence of…

In a recent article, the authors constructed a six-parameter family of highly connected 7-manifolds which admit an SO(3)-invariant metric of non-negative sectional curvature. Each member of this family is the total space of a Seifert…

Let $(M^{n},g)$ be a closed, connected, oriented, $C^{\infty}$, Riemannian, n-manifold with a transversely oriented foliation $\boldkey F$. We show that if $\lbrace X,Y \rbrace$ are basic vector fields, the leaf component of $[X,Y]$,…

We construct an invariant of closed ${\rm spin}^c$ 4-manifolds using families of Seiberg-Witten equations. This invariant is formulated as a cohomology class on a certain abstract simplicial complex consisting of embedded surfaces of a…

A new symmetry of $(1,0)$ supersymmetric non-linear $\sigma$-models in two dimensions with Fermi and mass sectors is introduced. It is a generalisation of the so-called special holonomy $W$-symmetry of Howe and Papadopoulos associated with…

To every real analytic Riemannian manifold M there is associated a complex structure on a neighborhood of the zero section in the real tangent bundle of M. This structure can be uniquely specified in several ways, and is referred to as a…

Noting that the complete lift of a Rimannian metric defined on a differentiable manifold is not 0-homogeneous on the fibers of the tangent bundle . In this paper we introduce a new lift which is 0-homogeneous. It determines on slit tangent…

The tangent bundle as a $4n$-manifold is equipped with an almost hypercomplex pseudo-Hermitian structure and it is characterized with respect to the relevant classifications. A number of 8-dimensional examples of the considered type of…

Complex geometric properties of the orbits of a non-compact real form $G_0$ in a flag manifold $Z=G/Q$ of a complex semi-simple groups $G=G_0^\mathbb C$ are studied. Schubert varieties are used to construct a complex submanifold with…

We classify orbifold geometries which can be interpreted as moduli spaces of four-dimensional $\mathcal{N}\geq 3$ superconformal field theories up to rank 2 (complex dimension 6). The large majority of the geometries we find correspond to…

For each integer n\ge 2, we construct an irreducible, smooth, complex projective variety M of dimension n, whose fundamental group has infinitely generated homology in degree n+1 and whose universal cover is a Stein manifold, homotopy…

It is known that there exist complex solvmanifolds $(\Gamma\backslash G,J)$ whose canonical bundle is trivialized by a holomorphic section which is not invariant under the action of $G$. The main goal of this article is to classify the…

We introduce the concept of non-Archimedean metrics attached to a transcendental pseudoeffective cohomology class on a compact K\"ahler manifold. This is obtained via extending the Ross-Witt Nystr\"om correspondence to the relative case,…

Motivated by generalized geometry, we discuss differential geometric structures on the total space $\mathfrak{T}M$ of the bundle $TM\oplus T^*M$, where $M$ is a differentiable manifold; $\mathfrak{T}M$ is called a big-tangent manifold. The…