Levi-Civita connection

The Levi-Civita connection is a mathematical tool used in differential geometry to describe how vectors change as they move along curved surfaces or spaces. Imagine walking along a curved hill; to keep pointing in the same direction relative to the surface, you need to adjust your orientation because of the hill’s shape. Similarly, the Levi-Civita connection provides a way to measure and compare how vectors are transported across curved spaces, ensuring this process respects the space’s geometry (being "torsion-free" and preserving lengths). It underpins much of the mathematics used in Einstein's theory of General Relativity.