Related papers: Pseudo-Riemannian foliations and their graphs

We investigate isometric immersions $f\colon M^n\to\R^{n+2}$, $n\geq 3$, of Riemannian manifolds into Euclidean space with codimension two that admit isometric deformations that preserve the metric of the Gauss map. In precise terms, the…

Notions of compatible and almost compatible pseudo-Riemannian metrics, which are motivated by the theory of compatible (local and nonlocal) Poisson structures of hydrodynamic type and generalize the notion of flat pencil of metrics, are…

We construct one dimensional foliations which are subfoliations of two dimensional foliations in 3-manifolds. The subfoliation is by quasigeodesics in each two dimensional leaf, but it is not funnel: not all quasigeodesics share a common…

We consider a generalization of Riemannian geometry that naturally arises in the framework of control theory. Let $X$ and $Y$ be two smooth vector fields on a two-dimensional manifold $M$. If $X$ and $Y$ are everywhere linearly independent,…

Definition of $({\cal T}_{a},{\cal T}_{b})$-pseudosymmetric semi-Riemannian manifold is given. $({\cal T}_{a},{\cal T}_{b})$-pseudosy mmetric $(N(k),\xi)$-semi-Riemannian manifolds are classified. Some results for ${\cal…

We show geodesic completeness of certain compact locally symmetric pseudo-Riemannian manifolds of signature $(2,n)$. Our model space $\mathbf{X}$ is a $1$-connected, indecomposable symmetric space of signature $(2,n)$, that admits a unique…

Differential calculus on discrete spaces is studied in the manner of non-commutative geometry by representing the differential calculus by an operator algebra on a suitable Krein space. The discrete analogue of a (pseudo-)Riemannian metric…

We prove the generalized Obata theorem on foliations. Let M be a complete Riemannian manifold with a foliation F of codimension $q>1$ and a bundle-like metric. Then $(M, F)$ is transversally isometric to the q-sphere of radius 1/c in…

We study Riemannian nilmanifolds associated with graphs. We prove that such a nilmanifold is geodesic orbit if and only if it is naturally reductive if and only if its defining graph is the disjoint union of complete graphs and the…

The author discusses in some detail the old definitions of the curvature tensors for rigged metrized distributions on manifolds given by Schouten, Wagner, and Solov'ev. To calculate the Solov'ev sectional and Ricci curvatures for…

Given a semi-Riemannian manifold $(M,\langle \cdot,\cdot\rangle_g),$ we use the transnormal functions defined on $M$ to reduce fully nonlinear first order PDEs of the form \[ F(x,u,\langle \nabla_g u, \nabla_g u \rangle_g) = 0,\qquad…

Given a (singular, codimension 1) holomorphic foliation F on a complex projective manifold X, we study the group PsAut(X, F) of pseudo-automorphisms of X which preserve F ; more precisely, we seek sufficient conditions for a finite index…

We generalize the following classical result of Fubini for pseudo-Riemannian metrics: if three essentially different metrics on $M^{n\ge 3}$ share the same unparametrized geodesics, and two of them (say, $g$ and $\bar g$) are strictly…

In previous papers, a fundamental affine method for studying homogeneous geodesics was developed. Using this method and elementary differential topology it was proved that any homogeneous affine manifold and in particular any homogeneous…

The unit sphere $\mathbb S^3$ can be identified with the unitary group SU(2). Under this identification the unit sphere can be considered as a non-commutative Lie group. The commutation relations for the vector fields of the corresponding…

In this article, we examine the geometry of a group of Fourier-integral operators, which is the central extension of $Diff(S^1)$ with a group of classical pseudo-differential operators of any order. Several subgroups are considered, and the…

We prove that there are only finitely many isoparametrically foliated closed connected Riemannian manifolds with bounded geometry, fixed dimension $n\neq5$, and finite fundamental group, up to foliated diffeomorphism. In addition, we…

We recall the notion of (vertical) translating solitons in a product of a semi-Riemannian manifold $(M,g)$ and the real line. Mainly, we restrict our attention to those which are the graph of a smooth function. When dealing with…

We prove that Riemannian foliations on complete contractible manifolds have a closed leaf, and that all leaves are closed if one closed leaf has a finitely generated fundamental group. Under additional topological or geometric assumptions…

In this article, relations between the root space decomposition of a Riemannian symmetric space of compact type and the root space decompositions of its totally geodesic submanifolds (symmetric subspaces) are described. These relations…