arXiv research

In this note, we give a proof of the famous theorem of M. Morse dealing with the cancellation of a pair of non-degenerate critical points of a smooth function. Our proof consists of a reduction to the one-dimensional case where the question becomes easy to answer.

New minimax theorem connects Bayesian and minimax regret in partial monitoring.

Study on hypersurfaces with minimized distance between rulings.

Study shows exponential sample complexity for stabilizing certain linear systems.

New proof shows certain 4D metrics are non-degenerate if curvature is negative definite.

Theorem shows generic metrics yield non-degenerate geodesic nets.

New move proves infinitely many non-conjugate braids for certain links.

Develops new Poisson structures for moduli spaces.

We classify all (locally) homogeneous Levi non-degenerate real hypersurfaces in $\mathbb{C}^3$ with symmetry algebra of dimension $\geq 6$.

We show that non-degenerate hyperquadrics in R^{n+2} admit no skew branes. Stated more traditionally, a compact codimension-one immersed submanifold of a non-degenerate hyperquadric of euclidean space must have parallel tangent spaces at two distinct points. Similar results have been proven by others, but (except for e…

The study resolves a conjecture about harmonic forms on compact manifolds.

Study on singularities of specific polynomial functions.

The abstract discusses transformations on statistical and semi-Weyl manifolds with torsion.

New findings on non-congruent curves with identical signatures.

Researchers create non-degenerate harmonic functions on n-dimensional space.

The paper identifies all flat CR Lie groups and their structures.

Classifies 3D non-degenerate left-symmetric algebras.

Paper studies invariant distributions of bi-Hamiltonian structures.

We prove that desingularizations of non degenerate Poincaré-Einstein metrics with A1 singularities remain non degenerate. In principle this enables a recursive procedure to desingularize the other Fuchsian singularities. We illustrate this procedure by the A2 case.

We study non-degenerate CR geometries of hypersurface type that are symmetric in the sense that, at each point, there is a CR transformation reversing the CR distribution at that point. We show that such geometries are either flat or homogeneous. We show that non-flat non-degenerate symmetric CR geometries of hypersurf…

Study non-degenerate anisocurved surfaces in homogeneous 3-manifolds.

Paper describes links of mixed polynomials with specific properties.

We consider differential operators on a supermanifold of dimension $1|1$. We define non-degenerate operators as those with an invertible top coefficient in the expansion in the "superderivative" $D$ (which is the square root of the shift generator, the partial derivative in an even variable, with the help of an odd ind…

The paper proves uniform stable radius and Milnor number equality for specific mappings.

We give complete classification of C^2-regular and non-degenerate projectively Anosov flows on three dimensional manifolds. More precisely, we prove that such a flow on a connected manifold must be either an Anosov flow or represented as a finite union of $T^2 \times [0,1]$-models.

New classification of complex hypersurfaces in 3D.

Let $M$ be a connected, non-compact $m$-dimensional Riemannian manifold. In this paper we consider smooth maps $φ: M \to \mathbb{R}^n$ with images inside a non-degenerate cone. Under quite general assumptions on $M$, we provide a lower bound for the width of the cone in terms of the energy and the tension of $φ$ and a …

In this paper we study lightlike surfaces of Minkowski 3- space such that they have degenerate or non-degenerate planar normal sections. We first show that every lightlike surface of Minkowski $3-$ space has degenerate planar normal sections. Then we study lightlike surfaces with non-degenerate planar normal sections a…

New insights into black hole horizons from asymptotic expansions.

Classification of homogeneous almost complex 4-manifolds with non-degenerate torsion bundle

We obtain an explicit parametrization of stationary discs glued to some Levi non-degenerate hypersurfaces. These discs form a family which is invariant under the action of biholomorphisms. We use this parametrization to construct a local circular representation of these hypersurfaces. As a corollary, we get the uniquen…

The paper develops methods for unconstrained optimization on Riemannian manifolds.

This paper introduces a new homology theory for Yang-Baxter solutions.

We study an infinite dimensional ASD moduli space over the cylinder. Our main result is the formula of its local mean dimension. A key ingredient of the argument is the notion of non-degenerate ASD connections. We develop its deformation theory and show that there exist sufficiently many non-degenerate ASD connections …

In this article, we demonstrate methods for the local removal and modification of complex tangents to embeddings of $S^3$ into $\mathbb{C}^3$. In particular, given any embedding of $S^3$ and a neighborhood of the complex tangents of the embedding, we show that there exists a ($C^0$-close) totally real embedding which a…

A mixed type surface is a connected regular surface in a Lorentzian 3-manifold with non-empty spacelike and timelike point sets. The induced metric of a mixed type surface is a signature-changing metric, and their lightlike points may be regarded as singular points of such metrics. In this paper, we investigate the beh…

After gluing foliated complex manifolds, we derive a preparation-like theorem for singularities of codimension one foliations and planar vector fields (in the real or complex setting). Without computation, we retrieve and improve results of Levinson-Moser for functions, Dufour-Zhitomirskii for non degenerate codimensio…

An almost complex manifolds $(M^4,J)$ of real dimension 4 with non-degenerate torsion bundle admit a double absolute parallelism and it is provided the classification of homogeneous $(M^4,J)$ having an associated non-solvable Lie algebra. We extend such a classification to the analysis of the manifolds having an associ…

We study the class of non-degenerate homogeneous structures of linear type in the pseudo-Kähler, para-Kähler, pseudo-quaternion Kähler and para-quaternion Kähler cases. We show that these structures characterize spaces of constant holomorphic, para-holomorphic, quaternion and para-quaternion sectional curvature respect…

A spacelike surface in four-dimensional Lorentz-Minkowski spacetime through the lightcone has a meaningful lightlike normal vector field $η$. Several sufficient assumptions on such a surface with non-degenerate $η$-second fundamental form are established to prove that it must be a totally umbilical round sphere. With t…

We study singularities of Gauss maps of fronts and give characterizations of types of singularities of Gauss maps by geometric properties of fronts which are related to behavior of bounded principal curvatures. Moreover, we investigate relation between a kind of boundedness of Gaussian curvatures near cuspidal edges an…

The tangent bundle of a Riemannian manifold (M,g) with non-degenerated g-natural metric G that admits a Killing vector field is investigated. Using Taylor's formula (TM,G) is decomposed into four classes that are investigated separately. The equivalence of the existence of Killing vector field on M and TM is proved. Ke…

A Kaehler-Nijenhuis manifold is a Kaehler manifold M, with metric g, complex structure J and Kaehler form F, endowed with a Nijenhuis tensor field A that is compatible with the Poisson stucture defined by F in the sense of the theory of Poisson-Nijenhuis structures. If this happens, and if either AJ=JA or AJ=-JA, M is …

Superintegrable systems on surfaces are classified geometrically.

There are two parts of this paper. First, we discovered an explicit formula for the complex Hessian of the weighted log-Bergman kernel on a parallelogram domain, and utilised this formula to give a new proof about the strict convexity of the Mabuchi functional along a smooth geodesic. Second, when a C^{1,1}-geodesic co…

Let LM be the semigroup of non-degenerate based loops with a fixed initial/final frame in a Riemannian manifold M of dimension at least three. We compare the topology of LM to that of the loop space Omega FTM on the bundle of frames in the tangent bundle of M. We show that Omega FTM is the group completion of LM, and p…

A class of codimension one foliations has been recently introduced by imposing a natural compatibility condition with a closed maximally non-degenerate 2-form. In this paper we study for such foliations the information captured by a Donaldson type submanifold. In particular we deduce that their leaf spaces are homeomor…

Superintegrable systems on curved manifolds found to have Hessian structures.