Metric geometry is a branch of mathematics that studies geometric properties and relationships using a concept called a metric. A metric defines the distance between points in a space, allowing mathematicians to analyze shapes, angles, and other geometric features in a precise way. This field includes various types of geometry, such as Euclidean geometry, which deals with flat surfaces, and non-Euclidean geometries, which explore curved spaces. In metric geometry, the focus is on how distances and angles behave under different transformations. This can involve studying geodesics, which are the shortest paths between points on a surface, and understanding how these concepts apply in various contexts, such as in topology or differential geometry.