We consider equilateral and equiangular $n$-polygonal chains in ${\mathbf{R}}^3$ with $n$ vertices, i.e. equilateral and equiangular spatial polygonal chains with $n$ vertices. Let $M_{\theta }(n)$ be the configuration space consisting of such polygonal chains with the bond angle $\theta$. In this article, we assume that $n=5$ and that $\theta$ satisfies $\frac{\pi }{2}\le \theta <\pi$. This bond angle condition gives that each polygonal chain in $M_{\theta }(5)$ is an equilateral and equiangular simple open or closed polygonal chain. The aim of this article is to study the level sets of the chain length function on $M_{\theta }(5)$ under the bond angle condition.

In the study of geodesics on surfaces, the subjects and methods are different for spheres, tori, other closed surfaces and non-compact surfaces. We make the method used to study geodesics on 2-tori applicable to surfaces with genus 2 or higher. Let $S$ be a torus with boundary embedded in an orientable geodesically complete Finsler surface $M$. We define a distance ${\delta }_S$ on $S$ in such a way that ${\delta }_S(p,q)$ is the minimum length of curves in $M$ from $p$ to $q$ homotopic to curves in $S$ with same endpoints $p,q\in S$. Those geodesics are minimal with respect to ${\delta }_S$ but not in $M$. We use this distance ${\delta }_S$ to study geodesics in $M$ homotopic to curves in $S$.

We introduce a Kähler-like, and a G-Kähler-like almost Hermitian metric. We prove that a Kähler-like and G-Kähler-like quasi-Kähler manifold is Kähler.

We study the integral-type operators between the weighted Lipschitz spaces on a tree. We characterized the boundedness and the compactness of the integral-type operators.

We show a necessary and sufficient condition for the Fujiki-Oka resolutions of Gorenstein abelian quotient singularities to be crepant in all dimensions by using Ashikaga’s continued fractions. Moreover, we prove that any three dimensional Gorenstein abelian quotient singularity possesses a crepant Fujiki-Oka resolution as a corollary. This alternative proof of existence needs only simple computations compared with the results ever known.