arXiv research

We introduce and discuss (local) symmetries of geometric structures. These symmetries generalize the classical (locally) symmetric spaces to various other geometries. Our main tools are homogeneous Cartan geometries and their explicit description. This allows us to describe the structure of symmetric geometric structur…

Let $Σ$ be a $C^3$ compact symmetric convex hypersurface in $\mathbf{R}^{8}$. For some special cases, we prove that when $Σ$ carries exactly four geometrically distinct closed characteristics, then all of them must be symmetric.

Geodesic completeness proven for all compact locally symmetric Lorentz manifolds.

By making use of the classification of real simple Lie algebra, we get the maximum of the squared length of restricted roots case by case, thus we get the upper bounds of sectional curvature for irreducible Riemannian symmetric spaces of compact type. As an application, we verify Sampson's conjecture in all cases for i…

Investigates locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.

There is a well developed theory of weakly symmetric Riemannian manifolds. Here it is shown that several results in the Riemannian case are also valid for weakly symmetric pseudo-Riemannian manifolds, but some require additional hypotheses. The topics discussed are homogeneity, geodesic completeness, the geodesic orbit…

Classification extended for reducible symmetric spaces.

As a difference with the positive-definite Riemannian case, in the Lorentzian case there exists proper second-order symmetric spacetimes, i.e., those with vanishing second covariant derivative of the Riemannian tensor ($R_{λμνρ;α;β}=0$) which are not locally symmetric ($R_{λμνρ;α}\neq 0$). In fact, they lie in the clas…

We study families of submanifolds in symmetric spaces of compact type arising as exponential images of s-orbits of variable radii. Special attention is given to the cases where the s-orbits are symmetric.

In this paper, we extend Su-Zhang's Cheeger-Mueller type theorem for symmetric bilinear torsions to manifolds with boundary in the case that the Riemannian metric and the non-degenerate symmetric bilinear form are of product structure near the boundary. Our result also extends Bruening-Ma's Cheeger-Mueller type theorem…

Study differential properties of matrix square roots in specific cases.

The notion of a $Γ$-symmetric space is a generalization of the classical notion of a symmetric space, where a general finite abelian group $Γ$ replaces the group $Z_2$. The case $Γ=\Z_k$ has also been studied, from the algebraic point of view by V.Kac \cite{VK} and from the point of view of the differential geometry by…

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

We study infinitesimal semi-simple extrinsic symmetric spaces and give a classification in the symplectic case.

The paper defines symmetric brackets for skew-symmetric algebroids with totally skew-symmetric torsion.

The study characterizes and verifies equivariant embeddings of symmetric Kählerian manifolds.

We investigate periodic diffeomorphisms of non-compact aspherical manifolds (and orbifolds) and describe a class of spaces that have no homotopically trivial periodic diffeomorphisms. Prominent examples are moduli spaces of curves and aspherical locally symmetric spaces with non-vanishing Euler characteristic. In the i…

New method of symmetrization applied to PDEs on spheres.

We present an approach to solvable pseudo-Riemannian symmetric spaces based on papers of M.Cahen, M.Parker and N.Wallach. Thereby we reproduce the classification of solvable symmetric triples of Lorentzian signature $(1,n-1)$ and complete the case of signature $(2,n-2)$. Moreover we discuss the topology of non-simply-c…

We characterize symmetric spaces without focal points by the equality case of general equalities between geometric quantities.

Paper introduces Deep Sets for Symmetric Elements (DSS) layers for learning sets of symmetric elements.

We characterize symmetric spaces of non-positive curvature by the equality case of general inequalities between geometric quantities

The study proves strong cosmic censorship violation for spherically symmetric dust clouds.

The article enumerates doubly symmetric diagrams for knots up to 18 crossings.

The study confirms that certain symmetric spaces are formal.

In symmetric cones, a non-empty locus satisfies the WDVV equation, generalizing previous results.

In this work we study riemannian metrics on flag manifolds adapted to the symmetries of these homogeneous nonsymmetric spaces. We first introduce the notion of riemannian $Γ$-symmetric space when $Γ$ is a general abelian finite group, the symmetric case corresponding to $Γ=\Z_2$. We describe and study all the riemannia…

We investigate existence and stability of rotationally symmetric critical immersions of variational problems of higher order which were considered by Nitsche.

Systematic prolongation for Killing two-tensors in symmetric spaces.

A geometry with parallel skew-symmetric torsion is a Riemannian manifold carrying a metric connection with parallel skew-symmetric torsion. Besides the trivial case of the Levi-Civita connection, geometries with non-vanishing parallel skew-symmetric torsion arise naturally in several geometric contexts, e.g. on natural…

Classifies Higgs and co-Higgs bundles on symmetric spaces.

We consider a class of overdetermined problems in rotationally symmetric spaces, which reduce to the classical Serrin's overdetermined problem in the case of the Euclidean space. We prove some general integral identities for rotationally symmetric spaces which imply a rigidity result in the case of the round sphere.

In a Riemannian manifold, the existence of a new connection is proved. In particular cases, this connection reduces to several symmetric, semi-symmetric and quarter-symmetric connections; even some of them are not introduced so far. We also find formula for curvature tensor of this new connection.

In this article, we study complete pseudo-Riemannian manifolds whose cone admits a parallel symmetric 2-tensorfield. The situation splits in three cases: nilpotent, decomposable or complex Riemannian. In the complex Riemannian and decomposable cases we provide a classification. In the nilpotent case, we are able to des…

Finite singular times for symmetric network curvature flow.

We define virtual immersions, as a generalization of isometric immersions in a pseudo-Riemannian vector space. We show that virtual immersions possess a second fundamental form, which is in general not symmetric. We prove that a manifold admits a virtual immersion with skew symmetric second fundamental form, if and onl…

In this paper we study homogeneous backgrounds of type IIB supergravity where the underlying geometry is that of a symmetric space. We determine which ten-dimensional lorentzian symmetric spaces (up to local isometry) admit such backgrounds and in about two thirds of the cases we determine fully their moduli space.

We formulate a conjectural Lefschetz formula for locally symmetric spaces of finite volume. The formula can be verified in the compact case and for Riemann surfaces.

The paper connects Riemannian Gaussian distributions to random matrix theory and diffusion kernels.

Proves gap rigidity theorem for Hermitian symmetric spaces.

We obtain Ricci flat Kähler metrics on complex symmetric spaces of rank two by using an explicit asymptotic model whose geometry at infinity is interpreted in the wonderful compactification of the symmetric space. We recover the metrics of Biquard-Gauduchon in the Hermitian case and obtain in addition several new metri…

The sinh-Gordon equation is solved on finite, symmetric graphs.

New insights into Einstein hypersurfaces in symmetric spaces.

In an earlier paper we developed the classification of weakly symmetric pseudo--riemannian manifolds $G/H$ where $G$ is a semisimple Lie group and $H$ is a reductive subgroup. We derived the classification from the cases where $G$ is compact. As a consequence we obtained the classification of semisimple weakly symmetri…

Paper proves new inequalities for Einstein-Maxwell data sets.

A relatively simple algebraic framework is given, in which all the compact symmetric spaces can be described and handled without distinguishing cases. We also give some applications and further results.

The paper classifies Landsberg spherically symmetric Finsler metrics in various dimensions.

We study Nijenhuis structures on Courant algebroids in terms of the canonical Poisson bracket on their symplectic realizations. We prove that the Nijenhuis torsion of a skew-symmetric endomorphism N of a Courant algebroid is skew-symmetric if the square of N is proportional to the identity, and only in this case when t…