On the Geometric features of the Causal definitions of Space-time

General Relativity puts forward the definitions of Causal Structures that are much needed to understand the entire structure of Space-time. These definitions arise due to the Pseudo-Riemannian nature of Space-time, and these are very important to write down a clear set of definitions that govern the Causal behaviour of Space-time. We first discuss the relation between two events – a set of Causal definitions that give us an idea of the relation between two intervals that are elements in the Pseudo-Riemannian manifold, labelled as events (all the points $p \in M$ are defined as events). We will discuss extensively of the geometric nature of thee definitions, which describe the nature of points on $M$. We talk of the light-cone structures, which we will use to understand and depict these Causal structures. We discuss in a geometric detail these definitions, after which we discuss about energy conditions. From there, we model Singularities, and how they can be predicted using the satisfaction of certain conditions by the Space-time. We then discuss the Raychaudari theorem, given by sir Amal Raychaudari, which was the first Singularity theorem that identified the modelling of Singularities. We then discuss the Pattern, Penrose and the Hawking Singularity theorem, and their structure, or the basis on which they are defined. We will then talk briefly of the Cauchy initial data problem, and we will talk of the implications of it. We will discuss the working of an initial data set, and then discuss how they can be mathematically understood. We will focus on an analysis of the Cauchy problem, and we will look through the various points that describe an initial data set on a manifold.