arXiv research

Extends Calabi operator to Riemannian locally symmetric spaces.

Computed formulas for curvature operators and Poincaré polynomials of symmetric 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.

Study properties of semi-symmetric Lorentzian spaces, foliated manifolds.

We give a method to calculate spectra of the square of the Rarita-Schwinger operator on compact symmetric spaces. According to Weitzenböck formulas, the operator can be written by the Laplace operator, which is the Casimir operator on compact symmetric spaces. Then we can obtain the spectra by using the Freudenthal's f…

Identifies Lorentzian locally symmetric spaces where Calabi operator suffices to determine Killing operator range.

The paper constructs hypersurfaces in symmetric space products.

Right inverse found for Cartan differential in rank-1 symmetric spaces.

We classify the connected pseudo-Riemannian manifolds of signature $(p,q)$ with $q\ge5$ 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.

Study on eigenvalues of Laplace operator on 1-forms for symmetric spaces.

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.

We give a formula for the first eigenvalue of the Dirac operator acting on spinor fields of a spin compact irreducible symmetric space $G/K$.

For any triple $(M^n, g, \nabla)$ 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.

In two previous papers, we started a study of the first eigenvalue of the Dirac operator on compact spin symmetric spaces, providing, for symmetric spaces of "inner" type, a formula giving this first eigenvalue in terms of the algebraic data of the groups involved. We conclude here that study by giving the explicit exp…

The paper establishes a majorization result for symmetric matrices.

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.

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.

The abstract discusses cohomologies and deformations of Rota-Baxter operators on Lie algebroids and Koszul-Vinberg structures.

Researchers compute the full spectrum of Laplace operator on distance spheres in symmetric spaces.

The article improves Beckner's inequality for axially symmetric functions on the n-dimensional sphere.

Study examines preservation of curvature-adaptedness during 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 $L^p$-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.

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.

Study on symmetric operators on non-compact manifolds, focusing on their index modulo 2.

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.

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.

Inverts rank m symmetric tensor fields using line integrals.

Study of spectral flow in symmetric Toeplitz operator families.

In this paper, first we give a notion for linear Weingarten spacelike hypersurfaces with $P+aH=b$ in a locally symmetric Lorentz space $L_{1}^{n+1}$. Furthermore, we study complete or compact linear Weingarten spacelike hypersurfaces in locally symmetric Lorentz spaces $L_{1}^{n+1}$ 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…

In this paper, we introduce a new commuting condition between the structure Jacobi operator and symmetric (1,1)-type tensor field $T$, that is, $R_ξφT=TR_ξφ$, 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 …

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 …

Formula for Z_2-valued index of symmetric operators on manifolds.

A field of endomorphisms $R$ is called a Nijenhuis operator if its Nijenhuis torsion vanishes. In this work we study a specific kind of singular points of $R$ called points of scalar type. We show that the tangent space at such points possesses a natural structure of a left-symmetric algebra (also known as pre-Lie or V…

Solves a challenging case of Nijenhuis operator linearization in 2D.

Proves inequality for special 3D shapes, generalizing to non-symmetric ones.

Let $M$ be an odd-dimensional Euclidean space endowed with a contact 1-form $α$. We investigate the space of symmetric contravariant tensor fields on $M$ as a module over the Lie algebra of contact vector fields, i.e. over the Lie subalgebra made up by those vector fields that preserve the contact structure. If we cons…