Riemannian geometry is a branch of mathematics that studies curved surfaces and spaces. It extends the concepts of Euclidean geometry, which deals with flat surfaces, to more complex shapes. In Riemannian geometry, distances and angles are measured using a special tool called a Riemannian metric, which allows for the analysis of curved spaces. This field has important applications in various areas, including theory of relativity, where it helps describe the curvature of spacetime. Riemannian geometry also plays a crucial role in differential geometry and has implications in physics, engineering, and computer science, particularly in understanding shapes and forms.