Higher Teichm{“u}ller theory studies representations of a surface group into a general Lie group that arise as deformation of the classical Teichm{“u}ller space. In this thesis, we focus on the Riemannian geometry for one family of Higher Teichm{“u}ller spaces that are Hitchin components. We study a Riemannian metric, called the pressure metric, in the Hitchin component H3(S) of surface group representations into PSL(3,R) and prove that the Hitchin parametrization provides geodesic coordinates at the Fuchsian locus for the pressure metric in H3(S). The proof is a combination of thermodynamic formalism and Higgs bundle theory. We compute first derivatives of the pressure metric by using Thermodynamic formalism and subshifts of finite type. We then study flat connections from Hitchin’s equations and their parallel transports by invoking a gauge-theoretic formula.