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We classify all smooth flat Riemannian metrics on the two-dimensional plane. In the complete case, it is well-known that these metrics are isometric to the Euclidean metric. In the incomplete case, there is an abundance of…

We derive the equations of motion of a test particle in the equatorial plane around a static and spherically symmetric wormhole influenced by a radiation field including the general relativistic Poynting-Robertson effect. From the analysis…

Consider a closed Riemannian $n$-manifold $M$ admitting a negatively curved Riemannian metric. We show that for every Riemannian metric on $M$ of sufficiently small volume, there is a point in the universal cover of $M$ such that the volume…

This paper is concerned with a covering problem of Euclidean space by a particular arrangement of cones that are not necessarily full and are allowed to overlap. The problem provides an equivalent geometric reformulation of the solvability…

We present an algorithm for producing Delaunay triangulations of manifolds. The algorithm can accommodate abstract manifolds that are not presented as submanifolds of Euclidean space. Given a set of sample points and an atlas on a compact…

The inclusion of the Weyl squared term in the gravitational action is one of the most simple, yet non trivial modifications to General Relativity at high energies. Nevertheless the study of the spherically-symmetric vacuum solutions of this…

A semi-Riemannian manifold is said to satisfy $R\ge K$ (or $R\le K$) if spacelike sectional curvatures are $\ge K$ and timelike ones are $\le K$ (or the reverse). Such spaces are abundant, as warped product constructions show; they include,…

The various "defect wormholes" developed by Klinkhamer have recently attracted considerable attention -- especially in view of the fact that the simplest example, the so-called "vacuum defect wormhole", was claimed to be an…

We study how coarse-graining procedure of an underlying UV-complete quantum gravity gives rise to a connected geometry. It has been shown, quantum entanglement plays a key role in the emergence of such a geometric structure, namely a smooth…

We reconsider the role of wormholes in the AdS/CFT correspondence. We focus on Euclidean wormholes that connect two asymptotically AdS or hyperbolic regions with $\mathbb{S}^1\times \mathbb{S}^{d-1}$ boundary. There is no solution to…

Fold maps are smooth maps at each singular point of which it is represented as the product map of a Morse function and the identity map. Round fold maps are, in short, such maps the sets of all singular points of which are embedded…

We construct wormholes supported by axion flux in the presence of a positive cosmological constant. The solutions describe compact, one-handle bodies colloquially known as kettlebell geometries. The wormholes are perturbatively stable, but…

We introduce variational approximations for curve evolutions in two-dimensional Riemannian manifolds that are conformally flat, i.e.\ conformally equivalent to the Euclidean space. Examples include the hyperbolic plane, the hyperbolic disk,…

Comparison theorems are foundational to our understanding of the geometric features implied by various curvature constraints. This paper considers manifolds with a positive lower bound on either scalar, 2-Ricci, or Ricci curvature, and…

Wormholes are non-trivial topological structures that arise as exact solutions to Einstein's field equations, theoretically connecting distinct regions of spacetime via a throat-like geometry. While static traversable wormholes necessarily…

We investigate Euclidean wormholes in Einstein gravity with a massless scalar field in de Sitter space. Euclidean wormholes are possible due to the analytic continuation of the time as well as complexification of fields, where we need to…

The AdS/CFT correspondence relates quantum entanglement between boundary Conformal Field Theories and geometric connections in the dual asymptotically Anti-de Sitter space-time. We consider entangled states in the n-fold tensor product of a…

A Riemannian metric on a manifold M induces a family of Riemannian metrics on the loop space LM depending on a Sobolev space parameter s. We compute the connection forms of these metrics and the higher symbols of their curvature forms,…

We construct a model for the nucleation of a wormhole within a Lorentzian spacetime by employing techniques from topological surgery and Morse theory. In our framework, a 0-surgery process describes the neighborhood of the nucleation point…

We establish a gluing theorem for solutions of a Yamabe problem for manifolds with boundary studied by Escobar in the 90's. Given two scalar-flat Riemannian manifolds whose boundary has zero mean curvature and sharing a submanifold $K$, we…