In this paper, we show the following splitting theorem: For a proper $\mathrm{CAT}(0)$ space $X$ with the geodesic extension property, if a group $\Gamma=G_{1}\times G_{2}$ acts geometrically (i.e., properly discontinuously and cocompactly by isometries) on $X$, then $X$ splits as a product $X_{1}\times X_{2}$ and there exist geometric actions of $G_1$ and some subgroup of finite index in $G_2$ on $X_1$ and $X_2$, respectively.

This paper deals with properties of the Cheeger-Gromoll metric $\overline{g}$ introduced in 1988 by Musso and Tricerri on the tangent bundle $TM$ associated to a given Riemannian metric $(M, g)$. One can find here essentially the two following results: 1. A classification of Killing vector fields on $(TM,\overline{g})$. 2. A generalization of a result of M. Sekizawa concerning the non rigidity of the Cheeger-Gromoll metric.

The main result of this paper is a general criterion for linearly unrelated sequences which does not depend on divisibility. A criterion for irrational sequences is presented as a consequence. Applications and several examples are included.

Let $C$ be a smooth plane algebraic curve of degree $n\geq 3$. We give the upper bound for the weights of points on $C$ and if $C$ has an involution, i.e., an automorphism of order 2, then we give the lower bound for the weights of fixed points of the involution on $C$. Furthermore, we obtain all the possible Weierstrass gap sequences and weights of fixed points of the involution for the case $n=5$ or 6.

This paper is devoted to some weighted pointwise estimate for solutions of two dimensional wave equations. It gives a simple proof to obtain the best order of the lower bound of the lifespan of classical solutions to nonlinear systems.