The Levi-Civita connection is a mathematical tool used in differential geometry to define how to differentiate vector fields on a curved space. It is a specific type of connection that is compatible with the Riemannian metric, meaning it preserves the lengths of vectors and angles between them as they are parallel transported along curves. This connection is unique because it is both torsion-free and metric-compatible. The torsion-free property ensures that the order of parallel transport does not affect the result, while the metric compatibility guarantees that the inner product of vectors remains constant during transport. Together, these properties make the Levi-Civita connection essential for studying the geometry of manifolds.