In 1912 Bieberbach proved that every compact flat Riemannian manifold M is finitely covered by a flat torus. More precisely, M has the form (F\G)/H where G is a group of translations of Euclidean space, F c G is a discrete subgroup, and H is a finite group of isometries of the space of right cosets F\G. For a proof see e.g. Wolf [18]. The condition that M has a flat Riemannian metric can be sep...

We show that the space of negatively curved metrics of a closed negatively curved Riemannian n-manifold, n ≥ 10, is highly non-connected. Section 0. Introduction. Let M be a closed smooth manifold. We denote by MET (M) the space of all smooth Riemannian metrics on M and we consider MET (M) with the smooth topology. Note that the space MET (M) is contractible. A subspace of metrics whose section...

A connected Riemannian manifold (M,g) is said to be isotropy irreducible if for each point p ∈ M the isotropy group Hp, i.e. all isometries of g fixing p, acts irreducibly on TpM via its isotropy representation. This class of manifolds is of great interest since they have a number of geometric properties which follow immediately from the definition. By Schur’s lemma the metric g is unique up to...

The flag curvature is a natural extension of the sectional curvature in Riemannian geometry, and the S-curvature is a non-Riemannian quantity which vanishes for Riemannian metrics. There are (incomplete) nonRiemannian Finsler metrics on an open subset in Rn with negative flag curvature and constant S-curvature. In this paper, we are going to show a global rigidity theorem that every Finsler met...

Conformally Stäckel manifolds can be characterized as the class of $n$-dimensional pseudo-Riemannian $(M,G)$ on which Hamilton–Jacobi equation $$ G(\nabla u, \nabla u) = 0 for null geodesics and Laplace $-\Delta\_G , \psi 0$ are solvable by R-separation variables. In particular case in metric has Riemannian signature, they provide explicit examples metrics admitting a set $n-1$ commuting confor...

in this paper we develop a natural generalization of schauder basis theory, we term operator-valued basis or simply ov-basis theory, using operator-algebraic methods. we prove several results for ov-basis concerning duality, orthogonality, biorthogonality and minimality. we prove that the operators of a dual ov-basis are continuous. we also dene the concepts of bessel, hilbert ov-basis and obt...

In this work the spaces of Riemannian metrics on a closed manifold M are studied. On the space M of all Riemannian metrics on M the various weak Riemannian structures are defined and the corresponding connections are studied. The space AM of associated metrics on a symplectic manifold M,ω is considered in more detail. A natural parametrization of the space AM is defined. It is shown, that AM is...

We show that every inner divisor of the operator-valued coordinate function, zIE, is a Blaschke-Potapov factor. also introduce notion “rational” function and then Δ two-sided rational if only it can be represented as finite product; this extends to functions well-known result proved by V.P. Potapov for matrix-valued functions.