arXiv research

Semi-Equivelar maps are generalizations of Archimedean solids to the surfaces other than 2-sphere. There are eight semi-equivelar maps of types $\{3^{3},4^{2}\}$, $\{3^{2},4,3,4\}$, $\{6,3,6,3\}$, $\{3^{4},6\}$, $\{4,8^{2}\}$, $\{3,12^{2}\}$, $\{4,6,12\}$, $\{6,4,3,4\}$ exist on the torus. In this article we show the e…

Homology of torus knots stabilizes to loop space homology.

New dg-algebras link graph colorings to sheaves.

Study pairs of subspaces with or without a common complement in Hilbert spaces.

This study reveals efficient finite-difference computation for gradient regularization in deep learning.

Let (M,g) be a compact Riemannian manifold of dimension n. For k \in {0,...,n}, we denote Gr_{k}(M) the set of compact, connected and oriented submanifolds of M of dimension k. This set is called the non-linear Grassmannian. In this article, we endow Gr_{k}(M) with a smooth Fréchet manifold structure and investigate it…

For a given manifold $M$ we consider the non-linear Grassmann manifold $Gr_n(M)$ of $n$-dimensional submanifolds in $M$. A closed $(n+2)$-form on $M$ gives rise to a closed 2-form on $Gr_n(M)$. If the original form was integral, the 2-form will be the curvature of a principal $S^1$-bundle over $Gr_n(M)$. Using this $S^…

Study on conjugate points in a $C^*$-algebra's Grassmann manifold.

We obtain a classification up to isomorphism of complex-analytic supermanifolds with underlying space $\mathbb{CP}^1$ of dimension $1|3$ with retract $(k,k,k)$, where $k\in \mathbb{Z}$. More precisely, we prove that classes of isomorphic complex-analytic supermanifolds of dimension $1|3$ with retract $(k,k,k)$ are in o…

Study shows limits of Fuchsian surfaces in hyperbolic 3-manifolds.

Let $Gr_k(\H^n)$ be the Grassmannian manifold of Quaternionic $k$-planes in $\H^n$ and let $γ^n_k\to Gr_k(\H^n)$ denote the Stiefel bundle of quaternionic $k$-frames in $\H^n$. Let $σ$ denote the first symplectic Pontrjagin form associated with the universal connection on $γ^n_k$. We show that every 4-form $ω$ on a smo…

We show that the algebra of functions on the Grassmann supergroup Gr$_q(1| 1)$ has a (graded) Hopf algebra structure related to GL$_q(1| 1)$.

FFR proposes a new theory of gravity without gravitational waves.

Investigates integrable systems derived from Grassmannian sixfolds in 4D.

The 'anholonomic frame' method (see gr-qc/0005025, gr-qc/0001060 and hep-th/0110250) is applied for constructing new classes of exact solutions of vacuum Einstein equations with off-diagonal metrics in 4D and 5D gravity. We examine several black tori solutions generated by anholonomic transforms with non-trivial topolo…

Einstein's philosophy uses differential identities to derive GR field equations.

The purpose of this erratum is to correct the proof of Theorem A.0.1 in the appendix to our article ``Hadamard spaces with isolated flats'' math.GR/0411232, which was jointly authored by Mohamad Hindawi, Hruska and Kleiner. In that appendix, many of the results of math.GR/0411232 about CAT(0) spaces with isolated flats…

In this survey paper, we outline the proofs of the rigidity results for simple, thick, hyperbolic P-manifolds found in our three earlier papers math.GR/0506518, math.GT/0410476, and math.GR/0409586. We discuss how the arguments change in the two, three, and higher dimensional settings. This paper was written for the 22…

Study non-existence of biconservative hypersurfaces in Minkowski spaces.

Integrable flows on the Grassmannians Gr(N-1,N+1) are defined by the requirement of closedness of the differential N-1 forms $Ω_{N-1}$ of rank N-1 naturally associated with Gr(N-1,N+1). Gauge-invariant parts of these flows, given by the systems of the N-1 quasi-linear differential equations, describe coisotropic deform…

Let a=(p_1^{q_1}, ..., p_r^{q_r}) be a partition and a'=({p_1'}^{q_1'}, >..., {p_r'}^{q_r'}) be its conjugate. We will prove that if q_i, q_i > 1 for all i, then any irreducible subvariety X of Gr(m,n) whose homology class is an integral multiple of the Schubert class [σ_a] of type a is a Schubert variety of type a.

This paper defines and proves properties of Floer homology for sutured manifolds.

Given a complex projective algebraic variety, write H(X) for its cohomology with complex coefficients and IH(X) for its Intersection cohomology. We first show that, under some fairly general conditions, the canonical map H(X)\to IH(X) is injective. Now let Gr = G((z))/G[[z]] be the loop Grassmannian for a complex semis…

Proves existence and uniqueness of rotating fluid bodies in GR to second order.

The paper finds commutator formulas for Gradient Ricci Shrinker metrics and applies them to linear stability.

We investigate dispersionless integrable systems in 3D associated with fourfolds in the Grassmannian Gr(3,5). Such systems appear in numerous applications in continuum mechanics, general relativity and differential geometry, and include such well-known examples as the dispersionless Kadomtsev-Petviashvili equation, the…

We analyse the definition of quasi-local energy in GR based on a Hamiltonian analysis of the Einstein-Hilbert action initiated by Brown-York. The role of the constraint equations, in particular the Hamiltonian constraint on the timelike boundary, neglected in previous studies, is emphasized here. We argue that a consis…

For any Lie groupoid we construct an analytic index morphism taking values in a modified $K-theory$ group which involves the convolution algebra of compactly supported smooth functions over the groupoid. The construction is performed by using the deformation algebra of smooth functions over the tangent groupoid constru…

In the present paper a generalized Kählerian space $\mathbb{G}\underset 1 {\mathbb{K}}{}_N$ of the first kind is considered, as a generalized Riemannian space $\mathbb{GR}_N$ with almost complex structure $F^h_i$, that is covariantly constant with respect to the first kind of covariant derivative. Using the non-symmetr…

The paper constructs foliations of minimal surfaces in negatively curved 3-manifolds.

We study holomorphic extensions of Matsuki orbits in complex Grassmannians.

In a previous paper Kobayashi and Rieck defined the growth rate of the tunnel number of a knot $K$, a knot invariant that measures the asymptotic behavior of the tunnel number under iterated connected sum of $K$. We denote the growth rate by $\mbox{gr}_t(K)$. In this paper we construct, for any $ε> 0$, a hyperbolic kno…

Classifies invariant differential operators on a specific geometric space.

New boundary conditions improve Hamiltonian analysis in GR.

Grötzsch's papers review progress in quasiconformal geometry.

We consider symplectic manifolds with Hamiltonian torus actions which are "almost but not quite completely integrable": the dimension of the torus is one less than half the dimension of the manifold. We provide a complete set of invariants for such spaces when they are "centered" and the moment map is proper. In partic…

Pattern recognition identifies Giant Radio Sources from NVSS catalog.

Machine learning classifies gravitational wave signals to test General Relativity.

A theorem connects two Willmore energies in 4D.

Maps from 2-planes to projective spaces using quaternions and octonions.

The paper studies a special Grassmannian space and shows it's an orbit of a unitary group.

General Relativity can be reformulated as a diffeomorphism invariant SU(2) gauge theory. A new action principle for this "pure connection" formulation of GR is described.

Minimal coloring number found for Z-colorable links.

Study automorphisms and real structures on a special super-Grassmannian.

The paper explores minimal coloring numbers for $\mathbb{Z}$-colorable links.

The paper shows links can be colored with fewer colors than previously thought.

Let $X$ be a closed hyperbolic surface and $λ, η$ be weighted geodesic multicurves which are short on X. We show that the iterated grafting along $λ$ and $η$ is close in the Teichmueller metric to grafting along a single multicurve which can be given explicitly in terms of $λ$ and $η$. Using this result, we study the h…

Study finds minimal coloring numbers for torus links using rack colorings.