Geometric properties of some classes of Riemannian almost-product manifolds

The aim of this paper is to study the geometric properties of the thirty-six classes of Riemannian almost-product manifolds that appear considering the algebraic properties of the covariant derivative of the tensor field defining the structure with respect to the Levi-Civita connexion.     

A Riemannian manifold with maximal L p spectrum

We give an example for a Riemannian manifold with maximal L ^p spectrum. Received: 16 February 2008     

Riemannian manifolds with two circulant structures

We consider a three-dimensional Riemannian manifold equipped with two circulant structures—a metric g and a structure q, which is an isometry with respect to g and the third power of q is minus identity. We discuss some curvature properties of this manifold, we give an example of such a manifold and find a condition for q to be parallel with respect to the Riemannian connection of g.     

Continuous maximal regularity on uniformly regular Riemannian manifolds

We establish continuous maximal regularity results for parabolic differential operators acting on sections of tensor bundles on uniformly regular Riemannian manifolds M. As an application, we show that solutions to the Yamabe flow on M instantaneously regularize and become real analytic in space and time. The regularity result is obtained by introducing a family of parameter-dependent diffeomorphisms acting on functions on M in conjunction with maximal regularity and the implicit function theorem.     

On the geodesic mobility of Riemannian spaces

It is proved that the degree of geodesic mobility of ann-dimensional Riemannian space (M, g) which is not a space of constant curvature can take only the valuesp=m(m+1)/2+l, wherem is the number of linearly independent concircular covector fields onM andl ranges from 1 to [(n+1−m)/3]; the brackets denote the integer part of a number. Thus the problem of finding all lacunas in the distribution of degrees of geodesic mobility is completely solved for this class of spaces. Translated fromMatematicheskie Zametki, Vol. 68, No. 4, pp 620–626, October, 2000.     

Clifford structures on Riemannian manifolds

We introduce the notion of even Clifford structures on Riemannian manifolds, which for rank r=2 and r=3 reduce to almost Hermitian and quaternion-Hermitian structures respectively. We give the complete classification of manifolds carrying parallel rank r even Clifford structures: Kähler, quaternion-Kähler and Riemannian products of quaternion-Kähler manifolds for r=2,3 and 4 respectively, several classes of 8-dimensional manifolds (for 5?r?8), families of real, complex and quaternionic Grassmannians (for r=8,6 and 5 respectively), and Rosenfeld?s elliptic projective planes OP², (C⊗O)P², (H⊗O)P² and (O⊗O)P², which are symmetric spaces associated to the exceptional simple Lie groups F₄, E₆, E₇ and E₈ (for r=9,10,12 and 16 respectively). As an application, we classify all Riemannian manifolds whose metric is bundle-like along the curvature constancy distribution, generalizing well-known results in Sasakian and 3-Sasakian geometry.     

Point leaf maximal singular Riemannian foliations in positive curvature

We generalize the notion of fixed point homogeneous isometric group actions to the context of singular Riemannian foliations. We find that in some cases, positively curved manifolds admitting these so-called point leaf maximal SRF's are diffeo/homeomorphic to compact rank one symmetric spaces. In all cases, manifolds admitting such foliations are cohomology CROSSes or finite quotients of them. Among non-simply connected manifolds, we find examples of such foliations which are non-homogeneous.     

Measurable Riemannian structures associated with strong local Dirichlet forms

We introduce Riemannian‐like structures associated with strong local Dirichlet forms on general state spaces. Such structures justify the principle that the pointwise index of the Dirichlet form represents the effective dimension of the virtual tangent space at each point. The concept of differentiations of functions is studied, and an application to stochastic analysis is presented.     

Toric moment mappings and Riemannian structures

Coadjoint orbits for the group SO(6) parametrize Riemannian G-reductions in six dimensions, and we use this correspondence to interpret symplectic fibrations between these orbits, and to analyse moment polytopes associated to the standard Hamiltonian torus action on the coadjoint orbits. The theory is then applied to describe so-called intrinsic torsion varieties of Riemannian structures on the Iwasawa manifold.     

Induced Riemannian structures on null hypersurfaces

Given a null hypersurface L of a Lorentzian manifold, we construct a Riemannian metric on it from a fixed transverse vector field ζ. We study the relationship between the ambient Lorentzian manifold, the Riemannian manifold and the vector field ζ. As an application, we prove some new results on null hypersurfaces, as well as known ones, using Riemannian techniques.     

Berezin kernels and maximal degenerate representations associated with Riemannian symmetric spaces of Hermitian type

We introduce the Berezin kernels for Riemannian symmetric spaces of Hermitian type by restricting the maximal degenerate representations of the corresponding noncompactly causal Lie groups. Bibliography: 20 titles.Published in Zapiski Nauchnykh Seminarov POMI, Vol. 292, 2002, pp. 11–21.This revised version was published online in April 2005 with a corrected cover date and article title.     

The mobility of Riemannian spaces with respect to conformal mappings onto Einstein spaces

We give an estimate of the first lacuna in the distribution of mobility degrees r of n-dimensional (pseudo-)Riemannian spaces with respect to conformal mappings onto Einstein spaces. We obtain a tensor characteristic of spaces which are not conformally flat and have r = n − 1, which is the maximum possible value. Thus, we have found maximum mobile nonconformally flat spaces with r = n − 1.