Riemannian Geometry is a branch of mathematics that studies curved surfaces and spaces. It generalizes the concepts of Euclidean Geometry, which deals with flat surfaces, to include shapes that can bend and stretch. This field is essential for understanding the geometry of the universe, as it provides tools to analyze the properties of curved spaces. One of the key ideas in Riemannian Geometry is the concept of a Riemannian metric, which allows for the measurement of distances and angles on curved surfaces. This metric helps mathematicians and physicists describe the geometry of various objects, including manifolds, which are spaces that locally resemble Euclidean space but can have complex global structures.