arXiv research

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

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

Maps with boundary definite fold points restrict manifold structure.

Considering a solid 3-dimensional Klein bottle and a collaring of its boundary, can we extend a generic $C^\infty$ non-singular function defined on the collaring to the full solid Klein bottle without critical points? We give a condition on the Reeb graph of the given function that is necessary and sufficient for the e…

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 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…

In this paper, we define the set of singular grid diagrams $\mathcal{SG}$ which provides a unified description for singular links, singular Legendrian links, singular transverse links, and singular braids. We also classify the complete set of all equivalence relations on $\mathcal{SG}$ which induce the bijection onto e…

In this article we study the validity of the Whitney $C^1$ extension property for horizontal curves in sub-Riemannian manifolds endowed with 1-jets that satisfy a first-order Taylor expansion compatibility condition. We first consider the equiregular case, where we show that the extension property holds true whenever a…

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…

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 construct infinite-dimensional families of non-singular static space times, solutions of the vacuum Einstein-Maxwell equations with a negative cosmological constant. The families include an infinite-dimensional family of solutions with the usual AdS conformal structure at conformal infinity.

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.

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…

Stability of Wasserstein spaces under various convergence types.

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 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 …

Generalising the results in arXiv:1612.00281, we construct infinite-dimensional families of non-singular stationary space times, solutions of Yang-Mills-Higgs-Einstein-Maxwell-Chern-Simons-dilaton-scalar field equations with a negative cosmological constant. The families include an infinite-dimensional family of soluti…

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 develop a new approach to geometric quantization using the theory of convergence of metric measure spaces. Given a family of Kähler polarizations converging to a non-singular real polarization on a prequantized symplectic manifold, we show the spectral convergence result of $\bar{\partial}$-Laplacians, as well as th…

We use an elliptic system of equations with complex coefficients for a set of complex-valued tensor fields as a tool to construct infinite-dimensional families of non-singular stationary black holes, real-valued Lorentzian solutions of the Einstein-Maxwell-dilaton-scalar fields-Yang-Mills-Higgs-Chern-Simons-$f(R)$ equa…

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…

We prove an excision theorem for the singular instanton Floer homology that allows the excision surfaces to intersect the singular locus. This is an extension of the non-singular excision theorem by Kronheimer and Mrowka and the genus-zero singular excision theorem by Street. We use the singular excision theorem to def…

Extends G-signature theorem to Witt G-pseudomanifolds.

Extends calculus to topological manifolds using generalized functions.

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.

Let $F$ be a non-singular foliation on the plane with all leaves being closed subsets, $H^{+}(F)$ be the group of homeomorphisms of the plane which maps leaves onto leaves endowed with compact open topology, and $H^{+}_{0}(F)$ be the identity path component of $H^{+}(F)$. The quotient $π_0 H^{+}(F) = H^{+}(F)/H^{+}_{0}…

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 …

For a non-singular real algebraic projective curve, topological restrictions on a closed motion of a simple real divisor in its linear equivalence class are found.

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, …

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…

The paper studies singularities of pedal curves of hyperbolic frontals.

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

We study the differential geometric properties of the manifold of non-singular symmetric real matrices endowed with the trace metric; in case of positive definite matrices we describe the full group of isometries

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

We show that every non-compact simple real Lie algebra not isomorphic to so(n,1) has a unique conjugacy class of parabolic subalgebras whose nilradical is of Heisenberg type, or non-singular, and give some applications.

We prove existence of large families of solutions of Einstein-complex scalar field equations with a negative cosmological constant, with a stationary or static metric and a time-periodic complex scalar field.

Let $G(n)={\rm Sp}(n,1)$ or ${\rm SU}(n,1)$. We classify conjugation orbits of generic pairs of loxodromic elements in $G(n)$. Such pairs, called `non-singular', were introduced by Gongopadhyay and Parsad for ${\rm SU}(3,1)$. We extend this notion and classify $G(n)$-conjugation orbits of such elements in arbitrary dim…

Study non-Weinstein Liouville geometry via hyperbolic dynamics, proving rigidity results.