arXiv research

A manifold is locally \emph{$k$-fold symmetric}, if for any point and any $k$-dimensional vector subspace tangent to this point there exists a local isometry such that this point is a fixed point and the differential of the isometry restricted to that $k$-dimensional vector subspace is minus the identity. We show that …

Geodesic completeness proven for all compact locally symmetric Lorentz manifolds.

The object of the present paper is to study locally $φ$-symmetric LP-Sasakian manifolds admitting semi-symmetric metric connection and obtain a necessary and sufficient condition for a locally $φ$-symmetric LP-Sasakian manifold with respect to semi-symmetric metric connection to be locally $φ$-symmetric LP-Sasakian man…

Positive simplicial volume implies locally symmetric space structure.

Compact holonomy groups found in symmetric spaces.

In this work we prove that every locally symmetric smooth submanifold gives rise to a naturally defined smooth submanifold of the space of symmetric matrices, called spectral manifold, consisting of all matrices whose ordered vector of eigenvalues belongs to the locally symmetric manifold. We also present an explicit f…

We use maximal periodic flats to show that on a finite volume irreducible locally symmetric manifold of dimension $\geq 3$, no metric $g$ has more symmetry than the locally symmetric metric. We also show that if $g$ is a finite volume metric that is not locally symmetric, then its lift to the universal cover has discre…

Study finds lower bounds on flat cycles in congruence covers of symmetric spaces.

Extends Calabi operator to Riemannian locally symmetric spaces.

In this note we compute low degree rational Pontryagin classes for every closed locally symmetric manifold of noncompact type. In particular, we answer the question: Which locally symmetric M have at least one nonzero Pontryagin class?

In this paper, we use the powerful tool Milnor bases to classify all the $3-$dimensional connected and locally symmetric Riemannian Lie Groups by solving system of polynomial equations of structure constants of each Lie algebra . Moreover, we showed that $E_0(2)$, is the only Lie group with locally symmetric left invar…

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…

In this article, we summarize the results on symmetric conformal geometries. We review the results following from the general theory of symmetric parabolic geometries and prove several new results for symmetric conformal geometries. In particular, we show that each symmetric conformal geometry is either locally flat or…

Systematic prolongation for Killing two-tensors in symmetric spaces.

In this paper, we show that the simplicial volume of Q-rank one locally symmetric spaces covered by the product of R-rank one symmetric spaces is strictly positive.

New spectral analysis on non-compact spaces.

The paper calculates minimum Dehn colors for knots using symmetric local biquandle cocycles.

We study the $L^p$-spectrum of the Laplace-Beltrami operator on certain complete locally symmetric spaces $M=Γ\backslash X$ with finite volume and arithmetic fundamental group $Γ$ whose universal covering $X$ is a symmetric space of non-compact type. We also show, how the obtained results for locally symmetric spaces c…

Local equivalence found between maximally symmetric rolling and flat Cartan distributions.

This is a final step in a local classification of pseudo-Riemannian manifolds with parallel Weyl tensor that are not conformally flat or locally symmetric.

Investigates locally symmetric polynomial metrics in Riemannian and Finslerian surfaces.

Here we survey on the growth of systoles of arithmetic locally symmetric spaces under the congruence covering and give simple proofs for the best possible constants of Gromov for several important classes of symmetric spaces.

A reflexion space is generalization of a symmetric space introduced by O. Loos. We generalize locally symmetric spaces to local reflexion spaces in the similar way. We investigate, when local reflexion spaces are equivalently given by a locally flat Cartan connection of certain type.

Arithmetic spaces' thin parts are negligible, impacting Betti numbers.

We introduce the notion of weak commensurabilty of arithmetic subgroups and relate it to the length equivalence and isospectrality of locally symmetric spaces. We prove many strong consequences of weak commensurabilty and derive from these many interesting results about isolength and isospectral locally symmetric space…

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…

The purpose of the present paper is to study the globally and locally $\varphi$-${\cal T}$-symmetric $\left( \varepsilon \right)$-para Sasakian manifold in dimension $3$. The globally $\varphi$-${\cal T}$-symmetric $3$-dimensional $\left( \varepsilon \right)$-para Sasakian manifold is either Einstein manifold or h…

Local-to-global principle for Morse actions on symmetric spaces.

In this paper, we provide a concrete interpretation of equivariant Reidemeister torsion and demonstrate that Bismut-Zhang's equivariant Cheeger-Müller theorem simplifies considerably when applied to locally symmetric spaces. In a companion paper, this allows us to extend recent results on torsion cohomology growth and …

In this paper we address the problem of studying those Kähler manifolds whose first two coefficients of the associated TYZ expansion vanish and we prove that for a locally Hermitian symmetric space this happens only in the flat case. We also prove that there exist nonflat locally Hermitian symmetric spaces where all th…

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…

We establish the proportionality principle between the Riemannian volume and locally finite simplicial volume for Q-rank 1 locally symmetric spaces covered by products of hyperbolic spaces, giving the first examples for manifolds whose cusp groups are not necessarily amenable. Also, we give a simple direct proof of the…

Lie foliations with symmetric leaves are smoothly conjugate to homogeneous ones.

For a closed locally symmetric space M=Γ\G/K and a representation of G we consider the push-forward of the fundamental class in the homology of the linear group and a related invariant in algebraic K-theory. We discuss the nontriviality of this invariant and we generalize the construction to cusped locally symmetric sp…

Arithmetic spaces simplified to simplicial complexes.

We list up all the possible local orbit types of hyperbolic or elliptic orbits for the isotropy representations of semisimple pseudo-Riemannian symmetric spaces. It is key to give a recipe to determine the local orbit types of hyperbolic principal orbits by using three kind of restricted root systems and Satake diagram…

We calculate the volume entropy of local Hermitian symmetric spaces of noncompact type in terms of its invariant $r$, $a$, $b$.

We study the Selberg zeta and the theta function associated to bundles over even-dimensional locally symmetric spaces of rank one.

In this article, we discuss which semisimple locally symmetric spaces admit an AHS--structure invariant to local symmetries. We classify them for all types of AHS--structures and determine possible equivalence classes of such AHS--structures.

Let (M,g) be an Einstein manifold of dimension n \geq 4 with nonnegative isotropic curvature. We show that (M,g) is locally symmetric.

Established concavity principle for curved spaces.

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.

We show that the results of Foulon (1997 an 2002) and Kim (2007) (independently, Deng and Hou (2007)) about the nonexistence of locally symmetric Finsler metrics of positive or negative flag curvature are in fact local.

We introduce a new family of affine metrics on a locally strictly convex surface $M$ in affine 4-space. Then, we define the symmetric and antisymmetric equiaffine planes associated with each metric. We show that if $M$ is immersed in a locally strictly convex hyperquadric, then the symmetric and the antisymmetric plane…

The holomorphic torsion of a compact locally symmetric manifold is expressed as a special value of a zeta function built out of geometric data (closed geodesics) of the manifold.

We construct compactifications of Riemannian locally symmetric spaces arising as quotients by Anosov representations. These compactifications are modeled on generalized Satake compactifications and, in certain cases, on maximal Satake compactifications. We deduce that these Riemannian locally symmetric spaces are topol…

In this paper, we classify three-locally-symmetric spaces for a connected, compact and simple Lie group. Furthermore, we give the classification of invariant Einstein metrics on these spaces.

The article contains a survey of results on length-commensurable and isospectral locally symmetric spaces and related problems in the theory of semi-simple algebraic groups.