Abstract: This habilitation thesis deals with the interactions between geometry of and analysis on smooth manifolds in different situations. Chapter 1 summarises all the results obtained in the next chapters. Chapter 2 deals with the spectrum of the Dirac operator of the Berger metrics on a $3$-dimensional space. Estimates on the smallest eigenvalues of twisted Dirac operators on the complex projective space are computed and their limiting-case discussed in chapter 3. Chapter 4 focuses on a purely geometric question of classifying those Kähler spin manifolds with imaginary Kählerian Killing spinors. In chapter 5, we address the Yamabe problem on globally hyperbolic spacetimes. Chapter 6 is concerned with locally covariant quantization of fields for a large class of differential operators on spacetimes.