The existence of special geometric structures on Riemannian and pseudo-Riemannian manifolds $(M,g)$ has been a central theme in geometry for nearly a century, beginning with the study of Einstein metrics ($\mathrm{Ric}(g)=sg$). Many such structures have since been investigated, including Kähler, Calabi-Yau, hyper-Kähler, $G_2$, and Sasaki.

Since the latter half of the 20th century, the study of these structures on (pseudo-)Riemannian manifolds has been closely tied to the holonomy group of the Levi-Civita connection, thanks to the work of Berger, Bryant, Salamon, and others. Subsequently, the existence of specific spinors—sections of the spinor bundle satisfying certain PDEs, such as being parallel with respect to the Levi-Civita connection—has been linked to the study of holonomy.

This talk will review the historical development of special geometric structures and introduce the theory of spinors, explaining their relationship in the Riemannian and pseudo-Riemannian setting. Known results on the existence of special spinors will be summarized. Furthermore, I will present a new embedding theorem for Killing spinors (joint work with D. Conti) and demonstrate its application to the invariant setting (joint work with D. Conti and F.A. Rossi). Time permitting, I will also discuss the extent to which the main ideas have been applied to special geometric structures and connections with torsion.