Riemannian submersions

A Riemannian submersion is a mathematical concept describing a smooth map between two curved spaces (called Riemannian manifolds) that preserves distances in certain directions. Imagine projecting a higher-dimensional object onto a lower-dimensional surface, like casting a shadow. This projection maintains the geometric structure along "horizontal" directions, while "vertical" directions (those lost in the projection) are orthogonal and can vary freely. Riemannian submersions help us understand how complex shapes can be related to simpler ones, preserving geometric properties along specific directions, and are fundamental in areas like differential geometry and geometric analysis.