The geometry of Heisenberg group and special curves in Heisenberg group

This master thesis consist of four parts.The first chapter, fundamental definitions and theorems used in our study are given.In second chapter, firstly manifold strucrure of $\mathbb{R}^n$, the group of all $n\times n$ matrices with real enries $M_n(\mathbb{R})$ and general linear group over the real numbers are examined. Afterwards, Lie groups structure of theese manifolds are given.Third chapter is original part of our study. In this chapter, firstly, $N_3,~\mathbb{H}$ and $\mathbb{H}^*$ Heisenberg groups and their properties are given, later group structures, manifold structures and Lie group structures of these groups was examined.The fourth chapter, the characterizations to be helix and slant helix of curves in Heisenberg group $\mathbb{H}^*$ are examined.