For an isometrically immersed submanifold, the spherical Gauss map is the induced immersion of the unit normal bundle into the unit tangent bundle. Compact rank one symmetric spaces have the distinguishing feature that their geodesics are closed with the same period, and so we can define the manifold of geodesics as the quotient of the unit tangent bundle by geodesic flow. Through this quotient we define the geodesic Gauss map to be the Lagrangian immersion given by the projection of the spherical Gauss map. In this thesis we establish relationships between the minimality of isometrically immersed submanifolds of the sphere and complex projective space and the minimality of the geodesic Gauss map with respect to the Kähler-Einstein metric on the manifold of geodesics. In particular, we establish that for an isometrically immersed holomorphic submanifold of complex projective space, its geodesic Gauss map is minimal Lagrangian if it has conformal shape form.