arXiv research

Researchers found infinite families of non-singular static spacetimes with negative cosmological constant.

New non-singular spacetimes found with negative cosmological constant.

We study global aspects of complete, non-singular asymptotically locally AdS spacetimes solving the vacuum Einstein equations whose conformal infinity is an arbitrary globally stationary spacetime. It is proved that any such solution which is asymptotically stationary to the past and future is itself globally stationar…

Study finds solutions for spacetimes with negative cosmological constant.

Study constructs infinite families of non-singular black hole solutions with a negative cosmological constant.

Classifies left invariant Kundt structures on 3D Lie groups.

The paper investigates the singularity and extendibility of inflationary spacetimes.

We show that the Big Bang singularity of the Friedmann-Lemaitre-Robertson-Walker model does not raise major problems to General Relativity. We prove a theorem showing that the Einstein equation can be written in a non-singular form, which allows the extension of the spacetime before the Big Bang. The physical interpret…

We investigate 3-dimensional globally hyperbolic AdS manifolds containing "particles", i.e., cone singularities of angles less than $2π$ along a time-like graph $Γ$. To each such space we associate a graph and a finite family of pairs of hyperbolic surfaces with cone singularities. We show that this data is sufficient …

The paper solves conditions for non-singular extensions of fold maps.

Generically, the set of points along which two non-singular vector fields on the three-sphere are positively (resp. negatively) collinear form a link. We prove that the two vector fields are homotopic if and only if the linking number of those links is zero. We use this criterion to give a new proof of a result of Yano…

A series of old and recent theoretical observations suggests that the quantization of gravity would be feasible, and some problems of Quantum Field Theory would go away if, somehow, the spacetime would undergo a dimensional reduction at high energy scales. But an identification of the deep mechanism causing this dimens…

A solution to the normalized Ricci flow is called non-singular if it exists for all time with uniformly bounded sectional curvature. By using the techniques developed by the present authors, we study the existence or non-existence of non-singular solutions of the normalized Ricci flow on 4-manifolds with non-trivial fu…

New families of non-singular geodesic orbit nilmanifolds discovered.

In this article, we shall investigate the relationship between the existence or non-existence of non-singular solutions to the normalized Ricci flow and smooth structures on closed 4-manifolds, where non-singular solutions to the normalized Ricci flow are solutions which exist for all time $t \in [0, \infty)$ with unif…

Maps with boundary definite fold points restrict manifold structure.

We give the general Kerr-de Sitter metric in arbitrary spacetime dimension D\ge 4, with the maximal number [(D-1)/2] of independent rotation parameters. We obtain the metric in Kerr-Schild form, where it is written as the sum of a de Sitter metric plus the square of a null geodesic vector, and in generalised Boyer-Lind…

The paper solves conditions for extending circle-valued Morse functions.

Quantum states in 4-manifolds linked to hyperbolic geometry.

In this note we study the distribution of real inflection points among the ovals of a real non-singular hyperbolic curve of even degree. Using Hilbert's method we show that for any integers $d$ and $r$ such that $4\leq r \leq 2d^2-2d$, there is a non-singular hyperbolic curve of degree $2d$ in $\mathbb R^2$ with exactl…

It is shown that the Kerr-Newman solution, representing charged and rotating stationary black holes, admits analytic extension at the singularity. This extension is obtained by using new coordinates, in which the metric tensor becomes smooth on the singularity ring. On the singularity, the metric is degenerale - its de…

Local coordinates for non-singular pairs in complex and quaternionic hyperbolic groups.

It is shown that there exist non-singular cubic surfaces in CP^3 containing 5 twistor lines. This is the maximum number of twistor fibres that a non-singular cubic can contain. Cubic surfaces in CP^3 with 5 twistor lines are classified up to transformations preserving the conformal structure of S^4.

The construction of topological index maps for equivariant families of Dirac operators requires factoring a general smooth map through maps of a very simple type: zero sections of vector bundles, open embeddings, and vector bundle projections. Roughly speaking, a normally non-singular map is a map together with such a …

New families of weakly symmetric nilmanifolds discovered.

Stability of Wasserstein spaces under various convergence types.

Study differential properties of symmetric real matrices with a specific metric.

Defines and calculates foliation homology from flows.

We calculate the first and the second variation formula for the sub-Riemannian area in three dimensional pseudo-hermitian manifolds. We consider general variations that can move the singular set of a C^2 surface and non-singular variation for C_H^2 surfaces. These formulas enable us to construct a stability operator fo…

We investigate 3-dimensional globally hyperbolic AdS manifolds containing "particles", i.e., cone singularities along a graph $Γ$. We impose physically relevant conditions on the cone singularities, e.g. positivity of mass (angle less than $2π$ on time-like singular segments). We construct examples of such manifolds, d…

Study magnetic fields on special Lie groups, proving non-existence of certain types.

A compact 4-dimensional manifold is a non-singular graph-manifold if it can be obtained by the glueing T^2-bundles over compact surfaces (with boundary) of negative Euler characteristics. If none of glueing diffeomorphisms respect the bundle structures, the graph-structure is called reduced. We prove that any homotopy …

We find an obstruction to the existence of non-singular solutions to the normalized Ricci flow on four-manifolds with $b^+=1$. By using this obstruction, we study the relationship between the existence or non-existence of non-singular solutions of the normalized Ricci flow and exotic smooth structures on the topologica…

Study finds conditions for existence of specific pseudo-Riemannian cobordisms.

We study bi-Hamiltonian systems of hydrodynamic type with non-singular (semisimple) non-local bi-Hamiltonian structures and prove that such systems of hydrodynamic type are diagonalizable. Moreover, we prove that for an arbitrary non-singular (semisimple) non-locally bi-Hamiltonian system of hydrodynamic type, there ex…

Research extends a function to a non-orientable 3-manifold without critical points.

Extends G-signature theorem to Witt G-pseudomanifolds.

New approach to geometric quantization for symplectic manifolds.

We design non-singular cloaks enabling objects to scatter waves like objects with smaller size and very different shapes. We consider the Schrodinger equation which is valid e.g. in the contexts of geometrical and quantum optics. More precisely, we introduce a generalized non-singular transformation for star domains, a…

In the first part of this paper, we consider smooth maps from a compact orientable 3-manifold without boundary to the 2-sphere. We give a geometric criterion to decide whether two given maps are homotopic, based on the sets of points where the maps are equal or antipodal. We extend this criterion to non-singular vector…

Algorithm finds characteristic maps over complex shapes.

For a real, non-singular, 2-step nilpotent Lie algebra $\mathfrak{n}$, the group \Aut(\mathfrak{n})/\Aut_0(\mathfrak{n})$, where$\Aut_0(\mathfrak{n})$ is the group of automorphisms which act trivially on the center, is the direct product of a compact group with the 1-dimensional group of dilations. Maximality of some …

In this paper we study non-singular solutions of Ricci flow on a closed manifold of dimension at least 4. Amongst others we prove that, if M is a closed 4-manifold on which the normalized Ricci flow exists for all time t>0 with uniformly bounded sectional curvature, then the Euler characteristic $χ(M)\ge 0$. Moreover, …

Seminar held at JINR, Dubna, May 15, 2012. In General Relativity, spacetime singularities raise a number of problems, both mathematical and physical. One can identify a class of singularities - with smooth but degenerate metric - which, under a set of conditions, allow us to define proper geometric invariants, and to w…

The paper studies singularities of pedal curves of hyperbolic frontals.

Study describes algebraic structure of foliated homeomorphisms.

The paper studies Gauss sums and their applications in algebra and topology.

The study reveals chaos in geometric objects embedded in higher dimensions.