Differential geometry is the mathematical discipline studying smooth manifolds and their curvature using differential and integral calculus, linear algebra, and multilinear algebra; a manifold is a topological space that is locally homeomorphic to Euclidean space, and a differentiable manifold additionally carries a smooth structure permitting calculus operations such as tangent vectors and tangent bundles. A diffeomorphism is a bijective, differentiable map between differentiable manifolds whose inverse is also differentiable, forming an isomorphism of smooth manifolds that preserves differentiable structure, distinguishing it from the weaker notion of homeomorphism (a continuous bijection with continuous inverse that preserves only topological structure). These concepts sit at the foundation of differential geometry within pure mathematics, connecting to Riemannian geometry, general relativity, and other branches concerned with the geometry of curved spaces.
Differential geometry is the mathematical discipline studying smooth manifolds and their curvature using differential and integral calculus, linear algebra, and multilinear algebra; a manifold is a t…