MSGSS
Metric-Affine Connections: A Friendly Introduction to Geometric Deformations
Metric-affine connection theory provides various mechanisms for measuring geometric deformations, with the curvature of the Levi-Civita connection being the most well-known, though not the only one. To explore this topic, we utilize the language of tensors and introduce them through an index-based approach without delving into their deeper geometric significance. Instead, we focus on defining these quantities based on their transformational properties. Using basic calculus, we develop the concepts of covariant derivatives, connections, and metrics, ultimately leading to the connection decomposition theorem and the fundamental tensorial quantities—non-metricity, curvature, and torsion—that describe geometric deformations.