In the joint paper of the author with K. P. Tod [J. Reine Angew. Math. 491, 183-198 (1997; Zbl 0876.53029)] they showed all local solutions of the Einstein-Weyl equations in three dimensions, where the background metric is homogeneous with unimodular isometry group. In particular, they proved that there are no solutions with Nil or Sol as background metric. In this note, these two special cases are presented.

Summary: It is proven that the Poisson algebra of a locally conformal symplectic manifold is integrable by making use of a convenient setting in global analysis. It is also observed that, contrary to the symplectic case, a unified approach to the compact and non-compact case is possible.

Summary: Here, a relation between integrals and variational multipliers of a system of second-order ordinary differential equations is studied. A simple necessary and sufficient local condition for the existence of a multiplier is given.

[For the entire collection see Zbl 0699.00032.] The paper deals with a special problem of gauge theory. In his previous paper [The invariance of Sobolev spaces over noncompact manifolds, Partial differential equations, Proc. Symp., Holzhaus/GDR 1988, Teubner- Texte Math. 112, 73-107 (1989; Zbl 0681.58011)], the author introduced the Sobolev completions ${\overline{𝒞}}_P^k$ of the space ${𝒞}_P$ of all G-connections on a G-principal fibre bundle P. In the present paper, under the assumption of bounded curvatures and their...

[For the entire collection see Zbl 0742.00067.]The author formulates several theorems about invariant orders in Lie groups (without proofs). The main theorem: a simply connected Lie group $G$ admits a continuous invariant order if and only if its Lie algebra $L\left(G\right)$ contains a pointed invariant cone. V. M. Gichev has proved this theorem for solvable simply connected Lie groups (1989). If $G$ is solvable and simply connected then all pointed invariant cones $W$ in $L\left(G\right)$ are global in $G$ (a Lie wedge $W\subset L\left(G\right)$ is said to...

In the paper under review, the author presents some results on the basis of the Nash-Gromov theory of isometric immersions and illustrates how the same results and ideas can be extended to other structures.

In 2005 Gilkey and Nikčević introduced complete $(p+2)$-curvature homogeneous pseudo-Riemannian manifolds of neutral signature $(3+2p,3+2p)$, which are $0$-modeled on an indecomposable symmetric space, but which are not $(p+3)$-curvature homogeneous. In this paper the authors continue their study of the same family of manifolds by examining their isometry groups and the isometry groups of their $k$-models.

The author gives a survey of the history of isospectral manifolds that are non-isometric discussing the work of Milnor, Vignéras, Sunada, and de Turck and Gordon. She describes the construction of continuous isospectral deformations as introduced by Gordon, Wilson, De Turck et al. She also discusses the construction of isospectral plane domains due to Gordon, Webb, and Wolpert. Some new examples of isospectral non-isometric manifolds are given.

A flag manifold of a compact semisimple Lie group $G$ is defined as a quotient $M=G/K$ where $K$ is the centralizer of a one-parameter subgroup $exp\left(tx\right)$ of $G$. Then $M$ can be identified with the adjoint orbit of $x$ in the Lie algebra $𝒢$ of $G$. Two flag manifolds $M=G/K$ and $M^{\text{'}}=G/K^{\text{'}}$ are equivalent if there exists an automorphism $\phi :G\to G$ such that $\phi \left(K\right)=K^{\text{'}}$ (equivalent manifolds need not be $G$-diffeomorphic since $\phi$ is not assumed to be inner). In this article, explicit formulas for decompositions of the isotropy representation for all flag manifolds...