  In this paper, we analyze the theory of meromorphic $(1,0)$-forms
$\omega\in\mathcal{M}\Omega^{(1,0)}(\mathbb{CP}^1).$ Hence, we show that on a
compact Riemann surface of genus $g=0,$ isomorphic to $\mathbb{CP}^1,$ every
non-constant meromorphic function $f:X\to\mathbb{CP}^1$ has as many zeros as
poles, where each is counted according to multiplicities. Such an analysis
gives rise to the following result. Invoking the Riemann-Roch theorem for a
compact Riemann $X$ with canonical divisor $K,$ it follows that $deg(f)=0$ for
any principal divisor $(f):=D$ on $X.$ More precisely,
$\ell(D)-\ell(K-D)=deg(D)+1=1$ or $\ell(D)-\ell(K-D)-1=0.$ Furthermore, for a
diffeomorphism $\eta:X\to\mathbb{CP}^1$ of a certain kind, a multistep program
is implemented to show $X$ is a compact algebraic variety of dimension one,
i.e. a non-singular projective variety. Hence, we adopt a group-theoretic
approach and provide a useful heuristic, that is, a set of technical conditions
to facilitate the algebro-geometric analysis of simply connected Riemann
surfaces $X.$
