arXiv research

Extends geometrical description of tensor manifolds in tree-based formats.

The paper defines and studies the geometric mean for tensors and its associated Riemannian geometry.

In the literature we see that after introducing a geometric structure by imposing some restrictions on Riemann-Christoffel curvature tensor, the same type structure given by imposing same restriction on other curvature tensors being studied. The main object of the present paper is to study the equivalency of various ge…

Tensor approach simplifies Euclidean space descriptions.

GETF efficiently decomposes large-scale Boolean tensors.

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 …

Geometrically, tensors of fixed rank form a minimal submanifold.

Study spectral functionals on manifolds with torsion.

Summary of tensor tomography proofs on manifolds with boundaries.

Equivariant networks improve geometric prediction without scalar approximations.

We show that every Kaehler algebraic curvature tensor is geometrically realizable by a Kaehler manifold of constant scalar curvature. We also show that every para-Kaehler algebraic curvature tensor is geometrically realizable by a para-Kaehler manifold of constant scalar curvature

We show how the theory of $\mathbb{Z}_2^n$ -manifolds - which are a non-trivial generalisation of supermanifolds - may be useful in a geometrical approach to mixed symmetry tensors such as the dual graviton. The geometric aspects of such tensor fields on both flat and curved space-times are discussed.

We present an algebraic investigation of generalized and equiaffine curvature tensors in a given pseudo-Euclidean vector space and study different orthogonal, irreducible decompositions in analogy to the known decomposition of algebraic curvature tensors. We apply the decomposition results to characterize geometric pro…

Derives energy-momentum tensor from Standard Model, examines energy conditions.

Classifies weakly Einstein curvature tensors in 4D Euclidean space.

We consider the hyperbolic geometric flow $\frac{\partial^2}{\partial t^2}g(t)=-2Ric_{g(t)}$ introduced by Kong and Liu [KL]. When the Riemannian metric evolve, then so does its curvature. Using the techniques and ideas of S.Brendle [Br,BS], we derive evolution equations for the Levi-Civita connection and the curvature…

Signature tensors uniquely identify ODE solutions.

In this paper we provide a \emph{global} investigation of the geometry of parallelizable manifolds (or absolute parallelism geometry) frequently used for application. We discuss the different linear connections and curvature tensors from a global point of view. We give an existence and uniqueness theorem for a remarkab…

Defines a new tensor related to special geometric spaces.

The covariant derivative of the Kähler form of an almost pseudo-Hermitian or of an almost para-Hermitian manifold satisfies certain algebraic relations. We show, conversely, that any 3-tensor which satisfies these algebraic relations can be realized geometrically.

Study on tautness tensor for Riemannian foliations.

Derdzinski and Shen's theorem on the restrictions posed by a Codazzi tensor on the Riemann tensor holds more generally when a Riemann-compatible tensor exists. Several properties are shown to remain valid in this broader setting. Riemann compatibility is equivalent to the Bianchi identity of the new "Codazzi deviation …

The Bakry-Emery tensor gives an analog of the Ricci tensor for a Riemannian manifold with a smooth measure. We show that some of the topological consequences of having a positive or nonnegative Ricci tensor are also valid for the Bakry-Emery tensor. We show that the Bakry-Emery tensor is nondecreasing under a Riemannia…

The ABP method is used to prove geometric inequalities for submanifolds and tensors.

Geometric analysis on real analytic manifolds using seminorms.

We employ the language of Cartan's geometry to present a model for studying vector spaces of Killing two-tensors defined in pseudo-Riemannian spaces of constant curvature under the action of the corresponding isometry group. We also discuss geometric properties of joint invariants of Killing two-tensors defined in the …

We work in both the complex and in the para-complex categories and examine (para)-Kähler Weyl structures in both the geometric and in the algebraic settings. The higher dimensional setting is quite restrictive. We show that any (para)-Kaehler Weyl algebraic curvature tensor is in fact Riemannian in dimension at least 6…

We give coordinate formula and geometric description of the curvature of the tensor product connection of linear connections on vector bundles with the same base manifold. We define the covariant differential of geometric fields of certain types with respect to a pair of a linear connection on a vector bundle and a lin…

This article provides a pedagogically oriented introduction to geometric (Clifford) calculus on pseudo-Riemannian manifolds. Unlike usual approaches to the topic, which rely on embedding the geometric algebra either within a tensor algebra or within a vector manifold framework, here we define geometric calculus directl…

In this paper, we deal with a generalization of the geometry of parallelizable manifolds, or the absolute parallelism (AP-) geometry, in the context of generalized Lagrange spaces. All geometric objects defined in this geometry are not only functions of the positional argument $x$, but also depend on the directional ar…

The paper examines geometric properties of a unique spacetime model.

In this paper the authors study the hyperbolic geometric flow on Riemann surfaces. This new nonlinear geometric evolution equation was recently introduced by the first two authors motivated by Einstein equation and Hamilton's Ricci flow. We prove that, for any given initial metric on ${\mathbb{R}}^{2}$ in certain class…

The paper studies Stein-Weiss operators on symmetric tensors, extending previous work.

Information geometry provides a geometric approach to families of statistical models. The key geometric structures are the Fisher quadratic form and the Amari-Chentsov tensor. In statistics, the notion of sufficient statistic expresses the criterion for passing from one model to another without loss of information. Thi…

A geometric construction for obtaining a prolongation of a connection to a connection of a bundle of connections is presented. This determines a natural extension of the notion of canonical energy-tensor which suits gauge and gravitational fields, and shares the main properties of the energy-tensor of a matter field in…

The Ricci tensor (Ric) is fundamental to Einstein's geometric theory of gravitation. The 3-dimensional Ric of a spacelike surface vanishes at the moment of time symmetry for vacuum spacetimes. The 4-dimensional Ric is the Einstein tensor for such spacetimes. More recently the Ric was used by Hamilton to define a non-li…

Paper introduces Tensor Gauge Flow Models for better data encoding.

Study shows no hidden symmetries in specific spacetime metrics.

Revisits stress-energy tensor in Finsler spacetimes, showing it's anisotropic.

The paper characterizes integrability of tensors on manifolds.

The geometric constructions are elaborated on (semi) Riemannian manifolds and vector bundles provided with nonintegrable distributions defining nonlinear connection structures induced canonically by metric tensors. Such spaces are called nonholonomic manifolds and described by two equivalent linear connections also ind…

The main goal of this paper is to study the geometric structures associated with the representation of tensors in subspace based formats. To do this we use a property of the so-called minimal subspaces which allows us to describe the tensor representation by means of a rooted tree. By using the tree structure and the d…

Study stability of Einstein manifolds with boundary.

The paper explores symmetries in Kähler manifolds using Ricci tensor properties.

Analyzes properties of stiffness tensors for elastic wave imaging.

New invariants found for mappings between non-symmetric affine spaces.

Paper bounds tensor decomposition's RLCT, aiding Bayesian inference.

Modified Gibbs-Helmholtz equation geometric models for thermodynamics.