A Riemannian manifold is a type of geometric space that combines the concepts of manifolds and Riemannian geometry. It is a smooth, curved surface that allows for the measurement of distances and angles. Each point on the manifold has a tangent space, which is equipped with a Riemannian metric that defines how distances are calculated locally. These manifolds are essential in various fields, including physics and mathematics, as they provide a framework for understanding curved spaces. For example, general relativity describes the universe as a Riemannian manifold, where the curvature of space-time is influenced by mass and energy.