arXiv research

The paper studies deformations of Kundt metrics using nil-Killing vector fields.

We show that, given an immortal solution to the Ricci flow on a closed manifold with uniformly bounded curvature and diameter, the Ricci tensor goes to zero as t goes to infinity. We also show that if there exists an immortal solution on a closed 3-dimensional manifold such that the product of the square of the diamete…

Study Wintgen ideal submanifolds in curved spaces with specific curvature conditions.

It is shown that a self-dual neutral Einstein four-manifold of Petrov type III, admitting a two-dimensional null parallel distribution compatible with the orientation, cannot be compact or locally homogeneous, and its maximum possible degree of mobility is 3. Diaz-Ramos, Garcia-Rio and Vazquez-Lorenzo found a general c…

The study classifies manifolds based on their geometric properties and invariants.

Calculates lower bounds for type III Reidemeister moves in link diagrams.

Proves the Kundt conjecture in arbitrary dimensions, confirming its validity.

Kähler-Ricci flow singularity type is independent of initial metric.

We study the number of Reidemeister type III moves using Fox n-colorings of knot diagrams.

Necessary and/or sufficient conditions are studied for the existence, uniqueness and holonomicity of bases in which on sufficiently general subsets of a differentiable manifold the components of derivations of the tensor algebra over it vanish. The linear connections and the equivalence principle are considered form th…

Formulates Index III lemma and Rauch III theorem with applications.

We prove uniform curvature estimates for homogeneous Ricci flows: For a solution defined on $[0,t]$ the norm of the curvature tensor at time $t$ is bounded by the maximum of $C(n)/t$ and $C(n) ( scal(g(t)) - scal(g(0)) )$. This is used to show that solutions with finite extinction time are Type I, immortal solutions ar…

In this paper, we study ruled surfaces and quadrics in the 3-dimensional Euclidean space which are of finite $III$-type, that is, they are of finite type, in the sense of B.-Y. Chen, with respect to the third fundamental form. We show that helicoids and spheres are the only ruled and quadric surfaces of finite $III$-ty…

In this paper, we introduce a monotonicity formula for the mean curvature flow which is related to self-expanders. Then we use the monotonicity to study the asymptotic behavior of Type III mean curvature flow on noncompact hypersurfaces.

We propose a restricted class of tensor network state, built from number-state preserving tensors, for supervised learning tasks. This class of tensor network is argued to be a natural choice for classifiers as (i) they map classical data to classical data, and thus preserve the interpretability of data under tensor tr…

In this paper, we study how to get the Ricci expanders from W+-functional through the heat kernel estimate of the conjugate heat equation to the type III singularity of Ricci flow. The Gaussian upper and lower bounds are established for the related heat kernel in accordance to the interesting work of Cao-Zhang for the …

We investigate some relations concerning the first and the second Beltrami operators corresponding to the fundamental forms I, II, III of a surface in the three-dimensional Euclidean space and we study surfaces which are of finite type in the sense of B.-Y. Chen with respect to the fundamental forms II and III.

The local structure of the manifolds named in the title is described. Although curvature homogeneous, they are not, in general, locally homogeneous. Not all of them are Ricci-flat, which answers an existence question about type III Jordan-Osserman metrics, raised by Diaz-Ramos, Garcia-Rio and Vazquez-Lorenzo (2006).

Study properties of surfaces with nonvanishing third fundamental form.

Classification of 3-symmetric spaces with Ricci solitons.

Fino and Kath determined all possible holonomy groups of seven-dimensional pseu\-do-Rie\-man\-nian manifolds contained in the exceptional, non-compact, simple Lie group $\mathrm{G}_2^*$ via the corresponding Lie algebras. They are distinguished by the dimension of their maximal semi-simple subrepresentation on the tang…

Study of special geometric structures on Lie groups.

We consider static spacetimes whose spatial part admits foliations with the extrinsic curvature tensor K_{ab}=0. There are two complementary cases when the gradient of the lapse function points 1) to the direction of foliation or 2) orthogonally to it. Case 1) gives generalization of metrics like Bertotti-Robinson or N…

The paper proves Liouville-type theorems on Hadamard manifolds.

We consider ruled and quadric surfaces in the 3-dimensional Euclidean space which are of coordinate finite type with respect to the third fundamental form $III$, i.e., their position vector $\boldsymbol{x}$ satisfies the relation $Δ^{III}\boldsymbol{x}=\varLambda \boldsymbol{x}$ where $\varLambda$ is a square matrix o…

The projective curvature tensor $P$ is invariant under a geodesic preserving transformation on a semi-Riemannian manifold. It is well known that $P$ is not a generalized curvature tensor and hence it possesses different geometric properties than other generalized curvature tensors. The main object of the present paper …

Study classifies 3D Einstein manifolds with cyclic Ricci tensor.

The paper characterizes integrability of tensors on manifolds.

A list of possible holonomy groups contained the exceptional, non-compact Lie group $\mathrm{G}_2^{*}$ was provided by Fino and Kath. The classification is due to the corresponding holonomy algebras and divided into Type I, II and III, depending on the dimension of the socle being 1,2 or 3, respectively. It was also sh…

Defines a new tensor related to special geometric spaces.

Paper develops a method for causal representation learning from irregular tensors.

On 5-dimensional almost contact B-metric manifolds, the form of any Kähler-type tensor (i.e. a tensor satisfying the properties of the curvature tensor of the Levi-Civita connection in the special class of the parallel structures on the manifold) is determined. The associated 1-forms are derived by the scalar curvature…

Constructs a representation of the string 2-group on a von Neumann algebra.

The paper studies curvature tensors and hypersurfaces in Kenmotsu type manifolds.

We construct a metrical framed structure on the tensor bundle of a Riemannian manifold equipped with a Cheeger-Gromoll type metric and by restricting this structure to the tensor sphere bundle, we obtain an almost metrical paracontact structure on the tensor sphere bundle. Moreover, we show that the tensor sphere bundl…

Systematic prolongation for Killing two-tensors in symmetric spaces.

In this article we study any 4-dimensional Riemannian manifold $(M,g)$ with harmonic curvature which admits a smooth nonzero solution $f$ to the following equation \begin{eqnarray} \label{0002bx} \nabla df = f(Rc -\frac{R}{n-1} g) + x Rc+ y(R) g. \end{eqnarray} where $Rc$ is the Ricci tensor of $g$, $x$ is a constant a…

Study of 4D flows with nilpotent symmetry, showing immortal solutions and blowdown limits.

The paper examines properties of $W$-curvature tensor in relativistic space-times.

We study the statistical limits of both detecting and estimating a rank-one deformation of a symmetric random Gaussian tensor. We establish upper and lower bounds on the critical signal-to-noise ratio, under a variety of priors for the planted vector: (i) a uniformly sampled unit vector, (ii) i.i.d. $\pm 1$ entries, an…

We show that three-dimensional homogeneous Ricci flow solutions that admit finite-volume quotients have long-time limits given by expanding solitons. We show that the same is true for a large class of four-dimensional homogeneous solutions. We give an extension of Hamilton's compactness theorem that does not assume a l…

An associated Nijenhuis tensor of endomorphisms in the tangent bundle is introduced. Special attention is paid to such tensors for an almost hypercomplex structure and the metric of Hermitian-Norden type. There are studied relations between the six associated Nijenhuis tensors as well as their vanishing. It is given a …

Paper classifies Einstein-type manifolds with parallel Ricci tensor.

Study analyzes Hotelling-type tensor deflation for spiked tensors, providing insights into signal and noise.

The very definition of an Einstein metric implies that all its geometry is encoded in the Weyl tensor. With this in mind, in this paper we derive higher-order Bochner type formulas for the Weyl tensor on a four dimensional Einstein manifold. In particular, we prove a second Bochner type formula which, formally, extends…

We study constant mean curvature Lorentzian hypersurfaces of $\mathbb{R}^{1,d+1}$ from the point of view of its Cauchy problem. We completely classify the spherically symmetric solutions, which include among them a manifold isometric to the de Sitter space of general relativity. We show that the spherically symmetric s…

We continue to develop the tensor-algebra approach to knot polynomials with the goal to present the story in elementary and comprehensible form. The previously reviewed description of Khovanov cohomologies for the gauge group of rank N-1=1 was based on the cut-and-join calculus of the planar cycles, which are involved …

We define a Weyl-type curvature tensor that provides a characterisation for Finsler metrics of constant flag curvature. When the Finsler metric reduces to a Riemannian metric, the Weyl-type curvature tensor reduces to the classic projective Weyl tensor. In the general case, the Weyl-type curvature tensor differs from t…