If you’re diving into the differential geometry of manifolds, the most "useful feature" is arguably the .
It is the only connection that is both torsion-free and metric-compatible . This means it preserves the lengths of vectors and the angles between them as you move them across the manifold. Differential Geometry of Manifolds
In short, it’s the "operating system" that allows you to perform standard calculus on a non-Euclidean space. If you’re diving into the differential geometry of