Related papers: Pseudo-Manifold Geometries with Applications

The study of comparison theorems in geometry has a rich history. In this paper, we establish a comparison theorem for polyhedra in 3-manifolds with nonnegative scalar curvature, answering affirmatively a dihedral rigidity conjecture by…

A Riemannian manifold $M$ is called weakly symmetric if any two points in $M$ can be interchanged by an isometry. The compact ones have been well understood, and the main remaining case is that of 2-step nilpotent Lie groups. We give a…

The elliptic 3-manifolds are the closed 3-manifolds that admit a Riemannian metric of constant positive curvature, that is, those that have finite fundamental group. The (Generalized) Smale Conjecture asserts that for any elliptic…

We take advantage of the principal bundle geometry of the space of connections to obtain general results on the presymplectic structure of two classes of (pure) gauge theories: invariant theories, and non-invariant theories satisfying two…

In this paper a bijective correspondence between superminimal surfaces of an oriented Riemannian $4$-manifold and particular Lagrangian submanifolds of the twistor space over the $4$-manifold is proven. More explicitly, for every…

The space-time geometry is considered to be a physical geometry, i.e. a geometry described completely by the world function. All geometrical concepts and geometric objects are taken from the proper Euclidean geometry. They are expressed via…

In the recent past, nested structures in Riemannian manifolds has been studied in the context of dimensionality reduction as an alternative to the popular principal geodesic analysis (PGA) technique, for example, the principal nested…

A submanifold of a pseudo-Riemannian manifold is said to have parallel mean curvature vector if the mean curvature vector field H is parallel as a section of the normal bundle. Submanifolds with parallel mean curvature vector are important…

Lagrange geometry is the geometry of the tensor field defined by the fiberwise Hessian of a non degenerate Lagrangian function on the total space of a tangent bundle. Finsler geometry is the geometrically most interesting case of Lagrange…

A Riemannian geometry of noncommutative n-dimensional surfaces is developed as a first step towards the construction of a consistent noncommutative gravitational theory. Historically, as well, Riemannian geometry was recognized to be the…

We discuss Sasakian-Einstein geometry under a quasi-regularity assumption. It is shown that the space of all quasi-regular Sasakian-Einstein orbifolds has a natural multiplication on it. Furthermore, necessary and sufficient conditions are…

This paper introduces the concept of supermanifolds, viewed as the super-analogues of classical manifolds. Instead of treating supermanifolds as sets of points, we adopt an algebraic-geometric perspective, emphasizing the algebra of…

A double disk bundle is any smooth closed manifold obtained as the union of the total spaces of two disk bundles, glued together along their common boundary. The Double Soul Conjecture asserts that a closed simply connected manifold…

The Stacey-Roberts lemma states that a surjective submersion between finite-dimensional manifolds gives rise to a submersion on infinite-dimensional manifolds of smooth mappings by pushforward. This result is foundational for many…

We generalize Bonahon's characterization of geometrically infinite torsion-free discrete subgroups of PSL(2, $\mathbb{C}$) to geometrically infinite discrete isometry subgroups in the case of rank 1 symmetric spaces, and, under the…

A Riemannian metric bundle G(M) is a fiber bundle over a smooth manifold M, whose fibers are the spaces of symmetric, positive-definite bilinear forms on the tangent spaces of M, which represent the Rieman?nian metrics. In this work, we aim…

Albeverio, Kondratiev, and R\"{o}ckner have introduced a type of differential geometry, which we call lifted geometry, for the configuration space $\Gamma_X$ of any manifold $X$. The name comes from the fact that various elements of the…

Let $R$ be a closed, oriented topological 4-manifold whose Euler characteristic and signature are denoted by $e$ and $\sigma$. We show that if $R$ has order two $\pi_1$, odd intersection form, and $2e + 3\sigma \geq 0$, then for all but…

We define a class of two dimensional surfaces conformally related to minimal surfaces in flat three dimensional geometries. By the utility of the metrics of such surfaces we give a construction of the metrics of $2 N$ dimensional Ricci flat…

We find necessary and sufficient conditions for a complete $n$-dimensional Riemannian manifold of finite volume, whose curvature tensor has nullity at least $n-2$, to be a geometric graph manifold. In the process, we show that Nomizu's…