Method calculates spectra of Rarita-Schwinger operator on symmetric spaces.
problem Calculating spectra of the Rarita-Schwinger operator on compact symmetric spaces.
method Using Weitzenböck formulas, Laplace operator, Casimir operator, Freudenthal's formula, and branching rules.
result Obtained spectra on the sphere, complex projective space, and quaternionic projective space.
Extends Calabi operator to Riemannian locally symmetric spaces.
problem Local integrability conditions on Riemannian locally symmetric spaces.
method Generalizes Calabi operator to Riemannian locally symmetric spaces.
result Generalised operator works in irreducible case and fails in products.
Study of Dirac operator eigenvalues on symmetric spaces.
problem Finding the first eigenvalue of the Dirac operator on compact spin symmetric spaces.
method Analyzing algebraic data of groups involved to derive formulas for eigenvalues.
result Explicit expression for the first eigenvalue of outer compact spin symmetric spaces.
Computed formulas for curvature operators and Poincaré polynomials of symmetric spaces.
problem Calculating curvature operators and Poincaré polynomials for symmetric spaces.
method Explicit formulas derived using quantum numbers and eigenvalue analysis.
result Maximum eigenvalue of curvature operators bounded by Einstein constant, with equality for Hermitian spaces.
We describe the shape of the symplectic Dirac operators on Hermitian symmetric spaces. For this, we consider these operators as families of operators that can be handled more easily than the original ones.
Classifies 3D non-degenerate left-symmetric algebras.
problem Classifying left-symmetric algebras in 3D.
method Using Nijenhuis geometry and algebraic independence of coefficients in characteristic polynomial.
result Classification of differentially non-degenerate LSA in dimension 3.
Study properties of semi-symmetric Lorentzian spaces, foliated manifolds.
problem Properties of semi-symmetric pseudo-Riemannian manifolds.
method Investigate foliated manifolds with Lorentzian metrics and analyze Ricci operator eigenvalues.
result Ricci operator has only real eigenvalues for Lorentzian metrics.
Identifies Lorentzian locally symmetric spaces where Calabi operator suffices to determine Killing operator range.
problem Determining when the Calabi operator can identify the range of the Killing operator in Lorentzian locally symmetric spaces.
method Developed criteria for a connection to be in the range of a connection, applied to the Killing connection.
result For indecomposable spaces, the Calabi operator suffices to identify the range of the Killing operator; for products, it fails.
The paper constructs hypersurfaces in symmetric space products.
problem Creating curvature-adapted hypersurfaces in symmetric space products.
method Constructing hypersurfaces using the product of symmetric spaces.
result Obtained many examples of curvature-adapted hypersurfaces.
Right inverse found for Cartan differential in rank-1 symmetric spaces.
problem Finding a right inverse for the Cartan differential in symmetric spaces.
method Integral operator approach to the Cartan differential on exact forms.
result Extension of Gauss linking integral to rank-1 symmetric spaces.
Study Nijenhuis operators and their linearization problem using left-symmetric algebras.
problem Linearization of Nijenhuis operators.
method Study points of scalar type, use left-symmetric algebras, classify 2D algebras.
result Complete classification of 2D real left-symmetric algebras.
We classify the connected pseudo-Riemannian manifolds of signature (p,q) with q≥5 so that at each point of M the skew-symmetric curvature operator has constant rank 2 and constant Jordan normal form on the set of spacelike 2 planes and so that the skew-symmetric curvature operator is not nilpotent for at least …
The paper studies Stein-Weiss operators on symmetric tensors, extending previous work.
problem Understanding Stein-Weiss operators on symmetric tensors of arbitrary rank.
method Analyzing the decomposition of tensor spaces into irreducible components and computing Weitzenbock formulas.
result Unified framework for second-order Stein-Weiss operators and tools for geometric analysis.
Study on eigenvalues of Laplace operator on 1-forms for symmetric spaces.
problem Investigating the first eigenvalue of the Laplace operator on 1-forms in compact inner symmetric spaces.
method Analyzing the Casimir eigenvalue of the highest root for the isotropy representation.
result The first eigenvalue of the Laplace operator on 1-forms is the Casimir eigenvalue of the highest root.
We associate to any Riemannian symmetric space (of finite or infinite dimension) a L∗-algebra, under the assumption that the curvature operator has a fixed sign. L∗-algebras are Lie algebras with a pleasant Hilbert space structure. The L∗-algebra that we construct is a complete local isomorphism invariant and …
Study on deformations of symmetric spaces using Jordan algebras.
problem Deformability of symmetric Einstein metrics on compact Lie algebras.
method Developed sandwich operators and quadratic Casimir operators for compact Lie algebras; calculated obstruction integrals from invariant polynomials; explored relation to simple Jordan algebras.
result Proved the nonlinear instability of most infinitesimally deformable irreducible compact symmetric spaces.
Estimates eigenvalues of Jacobi operator for harmonic spaces.
problem Estimating eigenvalues of Jacobi operator.
method Using density function of a harmonic space.
result Sharp estimates imply symmetric Osserman space.
For any triple (Mn,g,∇) consisting of a Riemannian manifold and a metric connection with skew-symmetric torsion we introduce an elliptic, second order operator Ω acting on spinor fields. In case of a reductive space and its canonical connection our construction yields the Casimir operator of the isometry gr…
Derives a formula for fermion dimensions in spherically symmetric monopole backgrounds.
problem Calculating the dimension of the plane-wave normalizable kernel for massless fermions in spherically symmetric monopole backgrounds.
method Derives a formula for the dimension of the plane-wave normalizable kernel of the Dirac operator for fermions of any representation of SU(N) in the presence of any spherically symmetric monopole background.
result Derives a formula for the dimension of the plane-wave normalizable kernel of the Dirac operator.
We give a formula for the first eigenvalue of the Dirac operator acting on spinor fields of a spin compact irreducible symmetric space G/K.
The paper establishes a majorization result for symmetric matrices.
problem Majorization of symmetric matrices under certain conditions.
method Using Garding-Dirichlet operator and properties of I-central operators. result A definitive inequality for symmetric matrices.
Optimal proof of finite small eigenvalues for specific geometric manifolds.
problem Proving finiteness of small eigenvalues for geometrically finite manifolds.
method Analyzing the spectrum of the Laplace operator on geometrically finite rank one locally symmetric manifolds.
result Optimal proof of finite small eigenvalues in a specific interval.
Let E be a natural operator associated to the curvature tensor of a pseudo-Riemannian manifold. This survey article studies when the spectrum, or more generally the real Jordan normal form, of E is constant on the natural domain of definition. It deals with results for the Jacobi operator, the higher order Jacobi opera…
Geometric operators link solutions on different spacetimes.
problem Comparing solutions on different globally hyperbolic manifolds.
method Intertwining operators preserving Hermitian forms.
result Existence of Hadamard states on globally hyperbolic manifolds.
We construct almost complex algebraic curvature tensors for pseudo Hermitian inner products whose skew-symmetric curvature operator has constant Jordan normal form on the set of non-degenerate complex lines.
The abstract discusses cohomologies and deformations of Rota-Baxter operators on Lie algebroids and Koszul-Vinberg structures.
problem Characterizing and studying deformations and cohomologies of relative Rota-Baxter operators.
method Constructing graded Lie algebras and studying their Maurer-Cartan elements, cohomology, and deformations.
result Homomorphisms between cohomology groups of relative Rota-Baxter operators and deformation cohomology groups of left-symmetric algebroids.
Researchers compute the full spectrum of Laplace operator on distance spheres in symmetric spaces.
problem Computing the full Laplace spectrum on distance spheres in symmetric spaces.
method Lie-theoretic methods to explicitly compute the spectrum.
result Unified formula for the full spectrum of Laplace operator on distance spheres in symmetric spaces of rank one.
The article improves Beckner's inequality for axially symmetric functions on the n-dimensional sphere.
problem Improving Beckner's inequality for axially symmetric functions on Sn. method Uniqueness and existence results for Q-curvature type equations with a Paneitz operator on Sn for axially symmetric functions. result Improved Beckner's inequality for axially symmetric functions on Sn. Study examines preservation of curvature-adaptedness during mean curvature flow.
problem Preservation of curvature-adaptedness during mean curvature flow.
method Investigates curvature-adaptedness in locally symmetric spaces.
result Curvature-adaptedness is preserved along mean curvature flow.
In this paper, we consider the symmetries of the Dirac operator derived from a connection with skew-symmetric torsion. We find that the generalized conformal Killing-Yano tensors give rise to symmetry operators of the massless Dirac equation, provided an explicitly given anomaly vanishes. We show that this gives rise t…
The notion of Lp-distributions is introduced on Riemannian symmetric spaces of noncompact type and their main properties are established. We use a geometric description for the topology of the space of test functions in terms of the Laplace-Beltrami operator. The techniques are based on a-priori estimates for ellipt…
New formula extracts full local information from ray transform data.
problem Determining symmetric tensor fields from ray transform data.
method Deriving explicit formula for Saint Venant operator.
result Explicit formula for extracting full local information.
In the present paper we show properties of a little-known Laplacian operator acting on symmetric tensors. This operator is an analogue of the well known Hodge-de Rham Laplacian which acts on exterior differential forms. Moreover, this operator admits the Weitzenböck decomposition and we study it using the analytical me…
Formula for Lefschetz numbers on locally symmetric spaces.
problem Calculating Lefschetz numbers on locally symmetric spaces.
method Introduced a new group C∗-algebra and used Hecke correspondences to study Lefschetz numbers. result Explicit formula for Lefschetz numbers of Hecke operators.
Study on symmetric operators on non-compact manifolds, focusing on their index modulo 2.
problem Investigating elliptic operators with a specific symmetry and their index modulo 2.
method Analysis of Callias-type operators on non-compact manifolds, establishing mod 2 versions of index theorems.
result Established mod 2 versions of the Gromov-Lawson relative index theorem, Callias index theorem, and Boutet de Monvel's index theorem for Toeplitz operators.
New CNN systems maintain invariant properties through combining symmetric operations or transformed inputs.
problem Maintaining invariant properties in CNNs for various transformations.
method Combining symmetric operations or transformed input vectors in parallel or averaging.
result Transformationally identical CNNs are mathematically equivalent and maintain invariant properties.
Part I. We prove a one-to-one correspondence between differential symmetry breaking operators for equivariant vector bundles over two homogeneous spaces and certain homomorphisms for representations of two Lie algebras, in connection with branching problems of the restriction of representations. We develop a new method…
Formula establishes determinant majorization for symmetric matrices.
problem Determining determinant majorization for symmetric matrices.
method Establishes determinant majorization formula for symmetric matrices using invariant Garding-Dirichlet polynomials.
result Formula F(A)N1≥det(A)n1 for symmetric matrices. It is shown that four-dimensional generalized symmetric spaces can be naturally equipped with some additional structures defined by means of their curvature operators. As an application, those structures are used to characterize generalized symmetric spaces.
Inverts rank m symmetric tensor fields using line integrals.
problem Recovering symmetric tensor fields from line integrals.
method Computes normal operator and presents inversion formula.
result Recovering rank m tensor fields from data (Nm0f,…,Nmmf). Study of spectral flow in symmetric Toeplitz operator families.
problem Understanding spectral flow in families of symmetric Toeplitz operators.
method Analog of Atiyah-Singer-Robbin-Salamon theorem for Z2-valued spectral flow. result Graded secondary spectral flow equals secondary index of a Callias-type operator.
In this paper, first we give a notion for linear Weingarten spacelike hypersurfaces with P+aH=b in a locally symmetric Lorentz space L1n+1. Furthermore, we study complete or compact linear Weingarten spacelike hypersurfaces in locally symmetric Lorentz spaces L1n+1 satisfying some curvature conditions…
Spectral clustering is a standard approach to label nodes on a graph by studying the (largest or lowest) eigenvalues of a symmetric real matrix such as e.g. the adjacency or the Laplacian. Recently, it has been argued that using instead a more complicated, non-symmetric and higher dimensional operator, related to the n…
Adapting the method of Andrews-Clutterbuck we prove an eigenvalue gap theorem for a class of non symmetric second order linear elliptic operators on a convex domain in euclidean space. The class of operators includes the Bakry-Emery laplacian with potential and any operator with second order term the laplacian whose fi…
This paper is the first of two papers constructing a calculus of pseudodifferential operators suitable for doing analysis on Q-rank 1 locally symmetric spaces and Riemannian manifolds generalizing these. This generalization is the interior of a manifold with boundary, where the boundary has the structure of a tower of …
In this paper, we introduce a new commuting condition between the structure Jacobi operator and symmetric (1,1)-type tensor field T, that is, RξφT=TRξφ, where T=A or T=S for Hopf hypersurfaces in complex hyperbolic two-plane Grassmannians. By using simultaneous diagonalzation for commuting symmetric operators…
Let G/K be a simply connected spin compact inner irreducible symmetric space, endowed with the metric induced by the Killing form of G sign-changed. We give a formula for the square of the first eigenvalue of the Dirac operator in terms of a root system of G. As an example of application, we give the list of the …
Formula for Z_2-valued index of symmetric operators on manifolds.
problem Index of symmetric operators on manifolds with boundary.
method Cohomological formula for Z_2-valued index.
result A formula for the Z_2-valued index of operators on manifolds.