Broder, Kyle2023-01-242023-01-24http://hdl.handle.net/1885/283940The present manuscript gives an account of the research undertaken from January 1st, 2019, to July 31st, 2022, under the supervision of Ben Andrews and Gang Tian. Outside of the Ph.D. requirement, the purpose of the present thesis is to provide (at least the author) with a useful reference on the curvature aspects of Hermitian manifolds. At the time of writing this, many texts exist concerning the geometry of K\"ahler manifolds, but not many of them have considered the general Hermitian category. The best reference at present appears to be Zheng's Complex Differential Geometry \cite{ZhengBook}, but the focus of this beautiful book is also not on Hermitian manifolds. In many respects, Hermitian (non-K\"ahler) differential geometry remains in its infancy. The presence of torsion in the natural connections which reside on the tangent bundle of a non-K\"ahler Hermitian manifold renders the subject formidable to outsiders, and this is undoubtedly exacerbated by the absence of books on the subject. The guiding narrative behind this thesis is to understand how the curvature of Hermitian metrics, which reside on a complex manifold, influences the complex geometry. The primary example of this type of investigation is in understanding when a sign on some curvature of a Hermitian metric force the manifold to be a known class (e.g., Kobayashi hyperbolic, Brody hyperbolic, Oka, homogeneous, etc.). This is the reason for titling the manuscript \textit{Complex Manifolds of Hyperbolic and Non-Hyperbolic-Type}. One of the central tools in studying hyperbolicity (and non-hyperbolicity) utilizing the curvature of Hermitian metrics is the Schwarz lemma (sometimes called the Yau--Schwarz lemma or Ahlfors--Schwarz lemma in this context). The new results established by the author primarily concern refinements and improvements on the Schwarz lemma (exhibited in $\S 2.5$ and $\S 2.6$), which appear in \cite{BroderSBC, BroderSBC2}. However, a better understanding of the Schwarz lemma (especially in the Hermitian category) requires an improved understanding of the holomorphic sectional curvature and Ricci curvature of a Hermitian metric. This, in turn, furnished some of the results in $\S 2.3$ and $\S 2.4$, which appear in \cite{BroderStanfield1, BroderStanfield2, BT1, BT2, BroderLA, BroderGraph, BroderRemarks}.en-AU202310.25911/PQY1-RT56