In this paper, we study the fundamental group of a certain class of globally hyperbolic Lorentzian manifolds with a positive curvature tensor. We prove that the fundamental group of lightlike geodesically complete parametrized Lorentzian products is finite under the conditions of a positive curvature tensor and the fib…
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48 results for “fundamental tensor”
The paper studies projectively and dually flat Finsler spaces with Randers changes.
problem Characterizing projectively and dually flat Finsler spaces with Randers changes.
method Analyzing Randers changes of special (α, β)-metrics, finding fundamental and inverse metric tensors, and establishing necessary conditions.
result Conditions for Randers changes of (α, β)-metrics to be projectively and locally dually flat.
Defines natural tensors for submanifolds of pseudo-Riemannian manifolds.
problem Characterizing tensors for submanifolds of pseudo-Riemannian manifolds.
method Constructs geodesic normal coordinates and expresses metric coefficients as polynomials in curvature and second fundamental form derivatives.
result Natural tensors are linear combinations of contractions of curvature and second fundamental form derivatives.
Calculates mass of hyperbolic manifolds using Ricci tensor and second fundamental form.
problem Evaluating the mass of asymptotically hyperbolic manifolds with noncompact boundaries.
method Uses Ricci tensor and second fundamental form via coordinates, similar to Miao-Tam's approach for asymptotically flat manifolds.
result Mass can be evaluated for these manifolds.
Paper proves existence of curves with specific geometric properties.
problem Existence of isometric immersions with prescribed second fundamental form.
method Introducing developments of curves with symmetric tensors and geometric construction.
result Existence of isometric immersions with prescribed second fundamental form.
Study of a 3D manifold with a circulant structure whose cube is the identity.
problem Characterizing a specific type of Riemannian manifold.
method Analyzing a tensor structure on a 3D manifold with a circulant property and its properties.
result An important characteristic identity for the fundamental tensor is derived.
In our previous paper (see this arxiv math.DG/0402171) for generic rank 2 vector distributions on n-dimensional manifold (n greater or equal to 5) we constructed a special differential invariant, the fundamental form. In the case n=5 this differential invariant has the same algebraic nature, as the covariant binary biq…
Conformally quasi-recurrent (CQR)_n pseudo-Riemannian manifolds are investigated, and several new results are obtained. It is shown that the Ricci tensor and the gradient of the fundamental vector are Weyl compatible tensors (the notion was introduced recently by the authors and applies to significative space-times), (…
Topology of spaces influences tensor fields on moduli spaces.
problem Understanding tensor fields on moduli spaces.
method Topological constructions.
result Tensor fields on moduli spaces can be influenced by the topology of the underlying space.
Generalizes O'Neill's equations to pseudo-Finsler submersions.
problem Extending Riemannian submersion equations to pseudo-Finsler geometry.
method Generalization of O'Neill's fundamental equations to pseudo-Finsler submersions and exploration of O'Neill tensors.
result Generalized fundamental equations for pseudo-Finsler submersions.
The classical Cartan's structural equations show in a compact way the relation between a connection and its curvature, and reveals their geometric interpretation in terms of moving frames. In order to study the mathematical properties of singularities, we need to study the geometry of manifolds endowed on the tangent b…
Study of hypersurfaces in Sol4_0 geometry, classifying parallel and totally umbilical types.
problem Classifying hypersurfaces in the Sol4_0 geometry.
method Analyzing hypersurfaces with Codazzi tensors and parallel second fundamental forms.
result Full classification of hypersurfaces in Sol4_0, including parallel and totally umbilical types.
We give an answer to a question posed recently by R.Bryant, namely we show that a compact 7-dimensional manifold equipped with a G2-structure with closed fundamental form is Einstein if and only if the Riemannian holonomy of the induced metric is contained in G2. This could be considered to be a G2 analogue of the Gold…
Study characterizes conformal boundaries of de Sitter spacetimes.
problem Characterize conformal infinity of asymptotically de Sitter spacetimes.
method Derive constraints relating stress-energy tensor to conformal geometric data using higher conformal fundamental forms.
result Constraints on stress-energy tensor relate to conformal geometric data.
Generalizes warped product submersion to conformal case.
problem Understanding angles preservation in submersions.
method Introduces conformal warped product submersion.
result Fundamental tensors derived for conformal submersion.
We show that if a complete Riemannian manifold supports a vector field such that the Ricci tensor plus the Lie derivative of the metric with respect to the vector field has a positive lower bound, then the fundamental group is finite. In particular, it follows that complete shrinking Ricci solitons and complete smooth …
Statistically Optimal and Computationally Efficient Low Rank Tensor Completion from Noisy Entriesstat.ML
New method for estimating low rank tensors from noisy data efficiently.
problem Estimating low rank tensors from noisy entries.
method Polynomial-time computable estimating procedure based on power iteration and spectral initialization.
result Achieves minimax optimal rates of convergence for noisy tensor completion.
We show that it is natural to consider the energy-momentum tensor associated with a spinor field as the second fundamental form of an isommetric immersion. In particular we give a generalization of the warped product construction over a Riemannian manifold leading to this interpretation. Special sections of the spinor …
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…
Study spectral learning for odeco tensors, addressing initialization bottlenecks.
problem Recovering orthogonally decomposable tensors under noise.
method Investigates perturbation bounds, non-convex optimization, and initialization strategies.
result Initialization is the main bottleneck for efficient algorithms.
The aim of the present paper is to provide an intrinsic investigation of projective changes in Finlser geometry, following the pullback formalism. Various known local results are generalized and other new intrinsic results are obtained. Nontrivial characterizations of projective changes are given. The fundamental proje…
Study proves no lightlike hypersurfaces exist in certain indefinite Sasakian manifolds.
problem Existence of lightlike hypersurfaces in indefinite Sasakian manifolds.
method Proved non-existence through properties of second fundamental forms and induced structural tensors.
result No lightlike hypersurfaces can have parallel or recurrent second fundamental forms or induced structural tensors.
Tensor networks help learn complex physical laws from data.
problem Identifying non-linear dynamical laws from complex physical systems.
method Tensor network parameterizations and rank-adaptive optimization.
result Optimal tensor network models can be learned from data.
The paper introduces a new method to characterize cosmological models using observer-based invariants.
problem Equivalence problem for cosmological models in four-dimensional gravity theories.
method Modified Cartan-Karlhede algorithm adapted to fundamental observers, including derivatives of the time-like vector field.
result A list of invariants that completely characterize cosmological models, independent of coordinates.
Study on specific types of Riemannian manifolds with detailed tensor analysis.
problem Characterizing and analyzing specific types of Riemannian manifolds.
method Detailed tensor analysis and characterization of manifolds in different classes.
result Characterization of manifolds with respect to a pair of tensors.
As is known, the Blaschke tensor (a symmetric covariant -tensor) is one of the fundamental Möbius invariants in the Möbius differential geometry of submanifolds in the unit sphere , and the eigenvalues of are referred to as the Blaschke eigenvalues. In this paper, we shall prove a classification…
The study proves non-existence theorems for Codazzi tensors on Riemannian manifolds.
problem Proving non-existence theorems for Codazzi tensors on Riemannian manifolds.
method Using theorems connecting manifold geometry and subharmonic functions.
result Several Liouville-type non-existence theorems for Codazzi tensors.
New tensors capture intrinsic embedding data of conformal hypersurfaces.
problem Classifying hypersurface invariants in conformal manifolds.
method Constructing curvatures and conformal fundamental forms.
result Finite family of tensors captures extrinsic embedding data.
On manifolds, we study the Energy-Momentum tensor associated with a spinor field. First, we give a spinorial Gauss type formula for oriented hypersurfaces of a manifold. Using the notion of generalized cylinders, we derive the variationnal formula for the Dirac operator under metric deformation and po…
New method for tensor recovery with fewer samples.
problem Recovering low-TT-rank tensors from few samples.
method Minimizing a weighted sum of nuclear norms of unfoldings.
result Significantly fewer samples required for recovery.
Study submanifolds with relative nullity in space forms using splitting tensor.
problem Characterize submanifolds with relative nullity in space forms.
method Use splitting tensor and Codazzi equation to express second fundamental form.
result Derive new strong consequences in hyperbolic and Euclidean spaces.
For a positive definite fundamental tensor all known examples of Osserman algebraic curvature tensors have a typical structure. They can be produced from a metric tensor and a finite set of skew-symmetric matrices which fulfil Clifford commutation relations. We show by means of Young symmetrizers and a theorem of S. A.…
The paper generalizes Riemann curvature for manifolds with discontinuous metrics.
problem Generalizing Riemann curvature for manifolds with discontinuous metrics.
method Proposes a generalized Riemann curvature tensor combining angle defects and jumps in second fundamental forms.
result The generalized curvature tensor approximates classical curvature for smooth approximations of metrics.
This paper studies how key tensor properties are inherited in subtensors of tensor train decompositions.
problem Theoretical development of property inheritance for subtensors in tensor train decompositions.
method Theoretical analysis of incoherence and condition number preservation, and tensor train rank preservation through fiber-wise sampling.
result Key tensor properties (incoherence and condition number) can be well preserved to subtensors formed via fiber-wise sampling.
The paper proves conditions for a hypersurface to be isoparametric.
problem Conditions for a hypersurface to be isoparametric.
method Analyzes a symmetric tensor field and its dual to show eigenvalues are constants.
result A closed hypersurface in is isoparametric under specific conditions.
The aim of the present paper is to provide an \emph{intrinsic} investigation of the properties of the most important geometric objects associated with the fundamental linear connections in Finsler geometry. We investigate intrinsically the most general relations concerning the torsion tensor fields and the curvature te…
Study spectral functionals on manifolds with torsion.
problem Extending classical results to geometries with torsion.
method Using Wodzicki residue, investigate spectral functionals of Dirac and Laplace-type operators.
result Local densities recover fundamental geometric tensors.
New tensors help determine if metrics are related to Poincaré-Einstein ones.
problem Determining when metrics on conformally compact manifolds are related to Poincaré-Einstein metrics.
method Developed new tensors and used conformal tractor calculus to analyze metrics.
result The vanishing of these new tensors is a necessary and sufficient condition for a metric to be related to a Poincaré-Einstein metric.
The paper develops tensor learning methods exploiting symmetries of tensor functions.
problem Efficiently handling tensors in various scientific contexts.
method Equivariant machine learning architectures exploiting orthogonal, Lorentz, and symplectic symmetries.
result Equivariant models outperform non-equivariant baselines in time series analysis.
Explicit formulas for the -components of the Riemannian curvature tensor on a manifold with a structure are given in terms of Ricci contractions. We define a conformally invariant Ricci-type tensor that determines the 27-dimensional part of the Weyl tensor and show that its vanishing on compact manifol…
Probabilistic Temporal Tensor Factorization (PTTF) is an effective algorithm to model the temporal tensor data. It leverages a time constraint to capture the evolving properties of tensor data. Nowadays the exploding dataset demands a large scale PTTF analysis, and a parallel solution is critical to accommodate the tre…
The fundamental tool in the classification of orthogonal coordinate systems in which the Hamilton-Jacobi and other prominent equations can be solved by a separation of variables are second order Killing tensors which satisfy the Nijenhuis integrability conditions. The latter are a system of three non-linear partial dif…
We construct an explicit categorification of the action of tangles on tensor powers of the fundamental representation of quantum sl(2).
Unified tensor network formalism for combining neural and symbolic AI.
problem Combining neural and symbolic AI approaches remains a challenge.
method Introduces a tensor network formalism capturing sparsity principles.
result Unified treatment identifies tensor network contractions as a fundamental inference class.
A novel Laplace-approximated Bayesian Tensor Network Kernel Machine (LA-TNKM) provides principled uncertainty estimates.
problem How to provide principled uncertainty estimates for tensor network kernel machines.
method Employing a linearized Laplace approximation for Bayesian inference.
result Consistently matches or surpasses Gaussian Processes and BNNs across diverse UCI regression benchmarks.
The Ahlfors Laplacian is applied to solve geometric and relativistic problems.
problem Solving geometric and relativistic problems using the Ahlfors Laplacian.
method Orthogonal decompositions and expansions of tensor components are used to study the Ahlfors Laplacian's applications.
result The Ahlfors Laplacian is applied to construct solutions of general relativistic constraint equations in vacuum.
ScaledGD algorithm estimates low-rank tensors efficiently from corrupted data.
problem Estimating meaningful information from corrupted tensor data.
method Scaled gradient descent (ScaledGD) algorithm with tailored spectral initializations.
result ScaledGD achieves linear convergence at a constant rate independent of condition number.
The study introduces new tensors for almost Finsler manifolds and analyzes their properties.
problem Defining and analyzing new types of Finsler manifolds.
method Introducing new Finsler manifolds, studying their properties, and deriving characteristic tensors.
result Characteristic tensors for almost Finsler manifolds have been generalized and their properties have been studied.