New Finsler metrics with constant flag curvature defined using Weyl-type curvature tensor.
problem Characterizing Finsler metrics with constant flag curvature.
method Defining a Weyl-type curvature tensor and constructing projectively related Finsler metrics.
result Construction of new families of Finsler metrics with constant flag curvature.
Finsler metrics of constant curvature characterized, with a Finslerian Beltrami Theorem.
problem Characterizing Finsler metrics of constant curvature.
method Defined a Weyl-type curvature tensor to characterize Finsler metrics of constant flag curvature.
result Provided a Finslerian version of Beltrami Theorem.
New invariants found for mappings between non-symmetric affine spaces.
problem Finding new invariants for mappings between non-symmetric affine spaces.
method Obtained invariants using factored deformation tensor and novel Weyl type invariants.
result Novel Weyl type invariants for mappings between non-symmetric affine spaces.
We study the structure group of a canonical algebraic curvature tensor built from a symmetric bilinear form, and show that in most cases it coincides with the isometry group of the symmetric form from which it is built. Our main result is that the structure group of the direct sum of such canonical algebraic curvature …
In this paper, by a new method we establish the Weyl-type asymptotic formula for the counting function of biharmonic Stekloff eigenvalues with Neumann boundary condition in a bounded domain of an n-dimensional Riemannian manifold.
We show that generalized plane wave manifolds are complete, strongly geodesically convex, Osserman, Szabo, and Ivanov-Petrova. We show their holonomy groups are nilpotent and that all the local Weyl scalar invariants of these manifolds vanish. We construct isometry invariants on certain families of these manifolds whic…
In this paper the projective curvature invariants of a complex Finsler space are obtained. By means of these invariants the notion of complex Douglas space is then defined. A special approach is devoted to obtain the equivalence conditions that a complex Finsler space should be Douglas. It is shown that any weakly Kähl…
The paper studies eigenvalues in gaps of the essential spectrum of a Bochner-Schrödinger operator.
problem Eigenvalue distribution in gaps of the essential spectrum of the Bochner-Schrödinger operator.
method Trace asymptotics formula and Weyl type asymptotic formula for eigenvalue counting function.
result The spectrum of Hp in the gap is discrete. We investigate the submanifold geometry of the orbits of Hermann actions on Riemannian symmetric spaces. After proving that the curvature and shape operators of these orbits commute, we calculate the eigenvalues of the shape operators in terms of the restricted roots. As applications, we get a formula for the volumes o…
We prove that in Riemannian manifolds the k-th Steklov eigenvalue on a domain and the square root of the k-th Laplacian eigenvalue on its boundary can be mutually controlled in terms of the maximum principal curvature of the boundary under sectional curvature conditions. As an application, we derive a Weyl-type upp…
Finite-dimensional spaces of biharmonic functions on manifolds are explored.
problem Characterizing biharmonic functions on open manifolds with nonnegative Ricci curvature.
method Analyzing bounded and polynomial growth biharmonic functions, deriving Weyl bounds, and studying fourth-order operators.
result Finite-dimensional spaces of biharmonic functions with polynomial growth are established.
We construct a new family of curvature homogeneous pseudo-Riemannian manifolds modeled on R3k+2 for integers k≥1. In contrast to previously known examples, the signature may be chosen to be (k+1+a,k+1+b) where a,b∈N⋃{0} and a+b=k. The structure group of the 0-model o…
The paper derives inequalities for eigenvalues and eigenfunction norms on manifolds.
problem Eigenvalue inequalities and eigenfunction norms on manifolds.
method Combining Milman's and Cheng-Li's work.
result Universal inequalities and upper bounds for eigenvalues and eigenfunction norms.
The geometry of the Lie algebroid generalized tangent bundle of a generalized Lie algebroid is developed. Formulas of Ricci type and identities of Cartan and Bianchi type are presented. Introducing the notion of geodesic of a mechanical (ρ,η)-system with respect to a (ρ,η)-spray, the Berwald (ρ,η)-…
We prove a Weyl-type fractal upper bound for the spectrum of the damped wave equation, on a negatively curved compact manifold. It is known that most of the eigenvalues have an imaginary part close to the average of the damping function. We count the number of eigenvalues in a given horizontal strip deviating from this…
Weyl-type theorems extended to Galilei and Carroll geometries.
problem Extending Weyl's theorem to non-relativistic and ultra-relativistic spacetimes.
method Defining and analyzing conformal and projective structures in Galilei and Carroll geometries.
result Torsion-free connections in Galilei and Carroll geometries are uniquely determined by their projective structures.
In the special case of S^1 invariant metrics on S^2, we find necessary and sufficient conditions for the existence of isometric embeddings into the canonical R^3, in other words: a Weyl type theorem with converse.
Estimates for spacelike hypersurfaces in de Sitter space.
problem Isometric embedding problem in de Sitter space.
method Obtained Weyl-type estimates and closedness in the space of embeddings.
result Weyl-type estimates and closedness in the space of embeddings.
The paper defines minimal norm tensors for curvature and divergence tensors, explaining Weyl and Cotten tensors.
problem Understanding curvature tensors and their minimal norm.
method Analyzing minimal norm tensors for third and fourth covariant tensors, including Riemannian curvature and divergence.
result Weyl tensor and Cotten tensor are identified as minimal norm tensors of Riemannian curvature and divergence tensors, respectively.
The paper examines properties of W-curvature tensor in relativistic space-times.
problem Investigating the properties and implications of the W-curvature tensor in relativistic space-times. method Analyzing the semi-symmetry and divergence properties of the energy-momentum tensor in relation to the W-curvature tensor. result Space-times with specific properties of the W-curvature tensor are classified as Einstein or Codazzi type. New tensors reveal full curvature structure from Riemann tensor.
problem Limited information from Ricci contraction of Riemann tensor.
method Contracting double dual of Riemann tensor to reveal full curvature.
result New tensors provide canonical parents of Einstein tensor.
Curvature tensors can always be matched to a metric tensor under certain conditions.
problem Sectionally positive curvature tensors and their relationship to metric tensors.
method Existence and uniqueness of a metric tensor gab such that Rabcdgbd=gacλ. result A metric tensor gab can be found for sectionally positive curvature tensors, and it is unique up to a constant factor. Proposes a new nonlocal curvature tensor concept.
problem Various nonlocal curvature concepts in literature.
method Generalizes classical curvature tensor representation and uses fractional differential operator analogies.
result Introduces a new nonlocal curvature tensor.
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…
Proves curvature tensor convergence for smoothable spaces.
problem Curvature tensor behavior in smoothable Alexandrov spaces.
method Weak convergence of curvature tensors in noncollapsing sequences.
result Proves convergence of curvature tensors in smoothable Alexandrov spaces.
The Riemann curvature tensor is a central mathematical tool in Einstein's theory of general relativity. Its related eigenproblem plays an important role in mathematics and physics. We extend M-eigenvalues for the elasticity tensor to the Riemann curvature tensor. The definition of M-eigenproblem of the Riemann curvatur…
Compatible tensors form a special Jordan algebra.
problem Understanding the algebraic structure of compatible tensors.
method Proving tensors form a Jordan algebra through symmetrized product properties.
result Riemann, Weyl, and curvature compatible tensors form a special Jordan algebra.
The paper studies quarter-symmetric connections on Hermitian and Kähler manifolds.
problem Examining quarter-symmetric connections on almost Hermitian and Kähler manifolds.
method Analyzing the curvature tensors and their properties with respect to quarter-symmetric connections.
result Constructed tensors that do not depend on the quarter-symmetric connection generator, including the Weyl projective curvature tensor.
We use the modified Riemannian extension of an affine surface to construct Bach flat manifolds. As all these examples are VSI (vanishing scalar invariants), we shall construct scalar invariants which are not of Weyl type to distinguish them. We illustrate this phenomena in the context of homogeneous affine surfaces.
In this article, we define a symmetric 2-tensor canonically associated to Q-curvature called J-tensor on any Riemannian manifold with dimension at least three. The relation between J-tensor and Q-curvature is precisely like Ricci tensor and scalar curvature. Thus it can be interpreted as a higher-order analogue of Ricc…
Completes the proof of curvature tensor existence for Jacobi operators.
problem Existence of curvature tensor for given Jacobi operators.
method Complete and accurate proof of the theorem, including a generalization to indefinite scalar product spaces.
result A complete proof of the existence of curvature tensor for given Jacobi operators, with a generalization.
Study on hypersurfaces with specific curvature conditions.
problem Characterizing hypersurfaces with certain curvature properties.
method Defined and analyzed the Opozda-Verstraelen affine curvature tensor for hypersurfaces.
result Conditions for pseudosymmetry types of hypersurfaces with specific curvature properties.
The paper defines and analyzes curvature tensors on super twisted product spaces.
problem Investigating curvature tensors on super twisted product spaces.
method Defined W2-curvature tensor, computed curvature tensors and Ricci tensors, and studied curvature flatness. result Mixed Ricci-flat super twisted product semi-Riemannian manifolds can be expressed as super warped product manifolds.
The study extends Jacobi-orthogonality to indefinite scalar product spaces.
problem Generalizing Jacobi-orthogonality to indefinite scalar product spaces.
method Comparing principles, investigating tensor relations, proving properties.
result Every quasi-Clifford tensor is Jacobi-orthogonal; certain tensors are Jacobi-dual or Osserman.
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 …
New characterization of Osserman tensors using Jacobi-orthogonality.
problem Characterizing Osserman tensors.
method Introducing Jacobi-orthogonality as a new potential characterization.
result Jacobi-orthogonal tensors are Osserman, and all known Osserman tensors are Jacobi-orthogonal.
Defines a new tensor related to special geometric spaces.
problem Understanding new curvature tensors in semi-Riemannian geometry.
method Defines a generalized curvature tensor using specific operations.
result The tensor is connected to quasi-Einstein, Roter, and Roter-type spaces.
Let R be an algebraic curvature tensor for a non-degenerate inner product of signature(p,q) where q>4. If π is a spacelike 2 plane, let R(π) be the associated skew-symmetric curvature operator. We classify the algebraic curvature tensors so R(-) has constant rank 2 and show these are geometrically realizable by hyp…
The paper studies Kropina metrics with a specific curvature property.
problem Characterizing Kropina metrics with isotropic scalar curvature.
method Tensor analysis to derive expressions and characterize metrics.
result Characterization of Kropina metrics with isotropic scalar curvature.
Local fractional derivatives affect Riemann curvature tensor to zero.
problem Investigating how local fractional derivatives influence the Riemann curvature tensor.
method Introduced a general local fractional derivative operator and defined a specific Riemannian metric tensor field.
result The Riemann curvature tensor of the new metric is identically zero, indicating local isometry to Euclidean space.
The paper examines geometric properties of a unique spacetime model.
problem Investigating the geometric properties of a point-like global monopole spacetime.
method Analyzing the spacetime's pseudosymmetry structures, energy-momentum tensor, and curvature properties.
result The point-like global monopole spacetime exhibits various pseudosymmetry structures and properties.
We use an isomorphism between the space of valence two Killing tensors on an n-dimensional constant sectional curvature manifold and the irreducible GL(n+1)-representation space of algebraic curvature tensors in order to translate the Nijenhuis integrability conditions for a Killing tensor into purely algebraic integra…
Develops Poisson and Dirac manifolds of compact types with applications.
problem Understanding Poisson and Dirac manifolds of compact types.
method Establishing structural results, local normal forms, canonical stratifications, and Weyl type resolutions.
result Every Poisson manifold of compact type is necessarily regular.
Study characterizes Finsler metrics with first integrals using specific curvature tensors.
problem Characterizing Finsler metrics with first integrals.
method Used χ-curvature and mean Berwald curvature to characterize the metrics. result Characterized a class of Finsler metrics admitting first integrals.
Survey on manifolds satisfying generalized Einstein conditions.
problem Characterizing semi-Riemannian manifolds under specific curvature conditions.
method Analyzing the difference tensor R.C-C.R expressed as linear combinations of Tachibana tensors.
result Recent results on manifolds and submanifolds satisfying generalized Einstein conditions.
Computes Finsler geometry curvature using anisotropic connections.
problem Computing curvature tensors in Finsler geometry.
method Using affine connections to compute curvature tensors without coordinates.
result Obtained Bianchi identities and comparison of curvature tensors.
Abstract Riemann curvature tensor defined on RCD spaces.
problem Defining Riemann curvature tensor on RCD spaces. method Distributional-like approach to introduce curvature tensor.
result Conjecture that RCD spaces are Alexandrov if curvature is bounded. It is well known that the curvature tensor of a pseudo-Riemannian manifold can be decomposed with respect to the pseudo-orthogonal group into the sum of the Weyl conformal curvature tensor, the traceless part of the Ricci tensor and of the scalar curvature. A similar decomposition with respect to the pseudo-unitary gro…