Analysis on manifolds

Analysis on manifolds is the study of functions, shapes, and equations defined on curved spaces that resemble smooth surfaces or higher-dimensional objects. Unlike flat spaces like regular graphs, manifolds can be curved, like Earth's surface. This field develops tools to understand how quantities change and behave on such shapes, including derivatives, integrals, and differential equations. It helps in fields like physics, engineering, and computer graphics by providing a rigorous way to analyze phenomena in complex, curved environments, enabling precise modeling and problem-solving beyond flat, Euclidean spaces.