A Riemannian manifold is a type of geometric space that allows for the measurement of distances and angles. It is defined by a smooth surface that has a Riemannian metric, which provides a way to calculate lengths of curves and the angles between them. This structure enables the study of curved spaces, extending the concepts of geometry beyond flat surfaces. These manifolds are essential in various fields, including general relativity, where they model the curvature of spacetime. They also play a significant role in differential geometry and topology, helping mathematicians understand complex shapes and their properties.