arXiv research

Ancient formula connects volume forms and infinitesimal square volumes in manifolds.

Hecke's theorem on different generalized to 3-manifolds.

New flag-no-square 4-manifolds discovered with unique triangulations.

We consider a special class of Finsler metrics --- square metrics which are defined by a Riemannian metric and a 1-form on a manifold. We show that an analogue of the Beltrami Theorem in Riemannian geometry is still true for square metrics in dimension $n\ge 3$, namely, an $n(\ge 3)$-dimensional square metric is locall…

The study examines harmonic symmetries on locally conformally Kähler manifolds, revealing properties of their kernels.

Investigates hypersurfaces in Finsler spaces with generalized square metrics.

Study homotopy types of 4-manifolds, finding decompositions and conditions for desuspension.

Paper proves uniqueness of minimal maps in curved spaces.

The paper identifies saddlepoints in unsupervised auto-encoding neural nets.

New perspective on Heegaard splittings using square complexes and combinatorial measurements.

Non-degeneracy of critical points proven for manifold's squared norm of second fundamental form.

B. Y. Chen establish the relationship between the Ricci curvature and the squared mean curvature for submanifolds of Riemannian space form with arbitrary codimension. In this paper, we generalize the relationship between the Ricci curvature and the squared norm of mean curvature vector for submanifolds of Bochner Kahle…

Defines a new Steenrod square for virtual links, linking to Khovanov-Lipshitz-Sarkar stable homotopy type.

We solve $\square_b$ on a class of non-compact 3-dimensional strongly pseudoconvex CR manifolds via a certain conformal equivalence. The idea is to make use of a related $\square_b$ operator on a compact 3-dimensional strongly pseudoconvex CR manifold, which we solve using a pseudodifferential calculus. The way we solv…

New geometric examples of involutions on Hilbert square K3 surfaces.

The paper broadens the class of manifolds where Dirac operator spectra are maximal.

New 4-manifold examples show necessary conditions for 4D Light Bulb Theorem.

The groups of differential characters of Cheeger and Simons admit a natural multiplicative structure. The map given by the squares of degree 2k differential characters reduces to a homomorphism of ordinary cohomology groups. We prove that the homomorphism factors through the Steenrod squaring operation of degree 2k. A …

In this paper, we prove that on any contact manifold, there exists an arbitrary C^{\infty}-small contactomorphism which does not admit a square root. In particular, there exists an arbitrary C^{\infty}-small contactomorphism which is not "autonomous". This result is the first step to study the topology of non-autonomou…

The eigenvalue problem for the square integrable solutions is studied usually for elliptic equations. In this note we consider such a problem for the hyperbolic Klein-Gordon equation on Lorentzian manifolds. The investigation could help to answer the question why elementary particles have a discrete mass spectrum. An i…

Estimates manifold dimension using local graph structure.

A classical result in Riemannian geometry states that the absolutely continuous curves into a (finite-dimensional) Riemannian manifold form an infinite-dimensional manifold. In the present paper this construction and related results are generalised to absolutely continuous curves with values in a strong Riemannian mani…

We prove the Chern-Weil formula for SU(n+1)-singular connections over the complement of an embedded oriented surface in smooth four manifolds. The expression of the representation of a number as a sum of nonvanishing squares is given in terms of the representations of a number as a sum of squares. Using the number theo…

Article generalizes open book construction for 5D contact pairs.

In this paper, the symplectic genus for any 2-dimensional class in a 4-manifold admitting a symplectic structure is introduced, and its relation with the minimal genus is studied. It is used to describe which classes in rational and irrational ruled manifolds are realized by connected symplectic surfaces. In particular…

New homotopy types defined for links in thickened surfaces with higher genus.

The study computes Bergman kernels and point process asymptotics on Kähler manifolds.

The square-peg problem is solved using configuration spaces and multijet transversality.

Study properties of bi-warped product submanifolds in specific geometric spaces.

The paper presents counterexamples to LS-category conjectures and constructs maps between manifolds.

We study the map degrees between quasitoric 4-manifolds. Our results rely on Theorems proved by Duan and Wang. We determine the set D (M, N) of all possible map degrees from M to N when M and N are certain quasitoric 4-manifolds. The obtained sets of integers are interesting, e. g. those representable as the sum of two…

Given a 3 manifold M with torus boundary and an ideal triangulation, Yoshida and Tillmann give different methods to construct surfaces embedded in M from ideal points of the deformation variety. Yoshida builds a surface from twisted squares whereas Tillmann produces a spun-normal surface. We investigate the relation be…

We present a proof due to Duistermaat that the gradient flow of the norm squared of the moment map defines a deformation retract of the appropriate piece of the manifold onto the zero level set of the moment map. Duistermaat's proof is an adaptation of Lojasiewicz's argument for analytic functions to functions which ar…

Functor connects symplectic and contact structures via cutting and blowups.

Study spider mechanism configuration spaces using squared distance function.

Characterizes algebraic squares of irreducible complex spinors in various dimensions.

We obtain a basic inequality involving the Laplacian of the warping function and the squared mean curvature of any warped product isometrically immersed in a Riemannian manifold without assuming any restriction on the Riemann curvature tensor of the ambient manifold. Applying this general theory, we obtain basic inequa…

The Kalinin effectivity is studied and applied to compactifications and Hilbert squares.

Recently, Hodgson and Kerckhoff found a small bound on Dehn surgered 3-manifolds from hyperbolic knots not admitting hyperbolic structures using deformations of hyperbolic cone-manifolds. They asked whether the area normalized meridian length squared of maximal tubular neighborhoods of the singular locus of the cone-ma…

Conformally recurrent pseudo-Riemannian manifolds of dimension n>4 are investigated. The Weyl tensor is represented as a Kulkarni-Nomizu product. If the square of the Weyl tensor is nonzero, a covariantly constant symmetric tensor is constructed, that is quadratic in the Weyl tensor. Then, by Grycak's theorem, the expl…

Study of pure spinors on neutral manifolds with applications to supersymmetric solutions.

Root Laplacian Eigenmaps help in spectral embedding of graphs.

New lattices in higher rank contain a fixed 3-manifold group with increasing systole.

In this paper we introduce slant Riemannian submersions from cosymplectic manifolds onto Riemannian manifolds. We obtain some results on slant Riemannian submersions of a cosymplectic manifolds. We also give examples and inequalities between the scalar curvature and squared mean curvature of fibres of such slant submer…

Smooth, globally PŁ functions are essentially nonlinear least-squares.

We show that for an isometric immersion of a complete Riemannian manifold into a Riemannian manifold with non-positive curvature, the norm of the mean curvature vector field is square integrable, then it is minimal. This is a partial affirmative answer of the B. Y. Chen's conjecture.

The configuration space $F_2 (M)$ of ordered pairs of distinct points in a manifold $M$, also known as the deleted square of $M$, is not a homotopy invariant of $M$: Longoni and Salvatore produced examples of homotopy equivalent lens spaces $M$ and $N$ of dimension three for which $F_2 (M)$ and $F_2 (N)$ are not homoto…

Geometrically refines Cramér-Rao bound using extrinsic manifold curvature.