We investigate the spaces of representations of surface groups into PSL(2, R), which extend the study of Teichmüller space and the mapping class group action on it. For a closed surface, by a classical result of Goldman, the Euler class provides a complete classification of the connected components of the character variety, and two of these components are identified with two copies of Teichmüller space. However, describing these components becomes more subtle when considering the relative character variety of type-preserving representations for punctured surfaces. In this talk, I will introduce two invariants of type-preserving representations that characterize the connected components. As an application, I will show the existence of totally hyperbolic representations, giving a negative answer to a question of Bowditch. I will also discuss the relation between the existence of such representations and the mapping class group action on the relative character variety.