A Finsler metric of a manifold or vector bundle is defined as a smooth assignment for each base point, a norm on each fiber space and thus the class of Finsler metrics contains Riemannian metrics as a special subclass. The geometry of complex Finsler manifold has been developed by [7]. In complex Finsler manifolds, the study of theory of curvatures has been an active field of research over past few decades. In the present article, our main purpose is to discuss some techniques of decomposition for the well known Cartan’s first curvature tensor $S_{jkh}^i$. Moreover, we attempted to establish few significant results that may produce vital connections between complex Finsler and complex Einstein’s manifolds. Also, by adopting the techniques of decomposition, various cases and conditions have been developed and their advantages in the study of theory of relativity & cosmology have been pursued.

We continue studying the properties and characterizations of $\gamma$-semi-$\theta$-closed sets, $\gamma$-$\mathrm{s}$-closed relative to a space $\mathrm{X}$ and locally $\gamma$-$\mathrm{s}$-regular spaces defined and discussed by B. Ahmad and S. Hussain in 2008 and S. Hussain and B. Ahmad in 2007 and 2009.

Let $R$ be a ring and $M$ a right $R$–module. In this paper we prove that if $M$ is weakly $\mathrm{F}$-supplemented, then every factor module and every $\mathrm{F}$-coclosed submodule of $M$ is again weakly $\mathrm{F}$-supplemented. In [5], it is shown that $Rad\left(M\right)$ has finite uniform dimension iff $M$ does not contain an infinite direct sum of nonzero small submodules. Here we replace $\mathrm{F}$-small submodules instead of small submodules (which is a weaker condition) and obtain the same result; i.e, we show that if $M$ does not contain an infinite direct sum of $\mathrm{F}$-small submodules, then $Rad\left(M\right)$ has finite uniform dimension.

This paper is motivated by the Cauty’s result [1] which states that if $X$ is a dendroid, then $2^X$ and $C\left(X\right)$ have the fixed point property. The main purpose of this paper is to study the fixed point property of the hyperspaces of arboroids. It is proved, using the inverse systems method and Cauty’s result, that if $X$ is an arboroid and $f:2^X\to 2^X$ is a mapping, then $f$ has the fixed point property. Similar theorem it is proved for $f:C\left(X\right)\to C\left(X\right)$.