Explore a unified framework for studying extremal curves on real Stiefel manifolds in this 49-minute lecture from the Thematic Programme on "Geometry beyond Riemann: Curvature and Rigidity" at the Erwin Schrödinger International Institute for Mathematics and Physics. Delve into a smooth one-parameter family of pseudo-Riemannian metrics on a product of orthogonal groups acting transitively on Stiefel manifolds. Discover Euler-Lagrange equations for a class of extremal curves, including geodesics with respect to different Riemannian metrics and smooth curves of constant geodesic curvature. Learn how specific parameter values in the family of pseudo-Riemannian metrics recover well-known metrics used in applied mathematics. This talk presents joint work with K. Hueper from the University of Wurzburg, Germany, and F. Silva Leite from the University of Coimbra, Portugal.