  The discovery of a square magnetic-skyrmion lattice in GdRu$_2$Si$_2$, with
the smallest so far found skyrmion diameter and without a geometrically
frustrated lattice, has attracted significant attention, particularly for
potential applications in memory devices and quantum computing. In this work,
we present a comprehensive study of surface and bulk electronic structures of
GdRu$_2$Si$_2$ by utilizing momentum-resolved photoemission (ARPES)
measurements and first-principles calculations. We show how the electronic
structure evolves during the antiferromagnetic transition when a peculiar
helical order of 4$f$ magnetic moments within the Gd layers sets in. A nice
agreement of the ARPES-derived electronic structure with the calculated one has
allowed us to characterize the features of the Fermi surface (FS), unveil the
nested region along the $k_z$ at the corner of the 3D FS, and reveal their
orbital compositions. Our findings suggest that the
Ruderman-Kittel-Kasuya-Yosida interaction plays a decisive role in stabilizing
the spiral-like order of Gd 4$f$ moments responsible for the skyrmion physics
in GdRu$_2$Si$_2$. Our results provide a deeper understanding of electronic and
magnetic properties of this material, which is crucial for predicting and
developing novel skyrmion-based devices.
