DifferentialGleichungssysteme

Differential geometry is a branch of mathematics that uses differential and integral calculus to study geometric properties and structures. It centers on differentiable manifolds, spaces that locally resemble Euclidean space, and on smooth maps between them. Fundamental objects include tangent spaces, vector fields, differential forms, and metrics that measure lengths and angles.

Core concepts include connections, which define how to differentiate vector fields along curves; the Levi-Civita connection, which is compatible with a Riemannian metric and torsion-free, is particularly important. Curvature measures how a space deviates from flatness, with the Riemann curvature tensor, the Ricci tensor, and scalar curvature representing related levels of detail. Geodesics generalize straight lines to curved spaces as curves that locally minimize distance or energy.

Techniques in differential geometry combine tensor calculus and exterior calculus to provide coordinate-free methods for analysis. Cartan’s structure equations relate curvature, torsion, and connections, and various formalisms (such as differential forms and frame fields) streamline computations on manifolds.

Applications of differential geometry span many fields. In physics, it underpins general relativity through pseudo-Riemannian geometry. In computer vision and graphics, it informs the analysis of surfaces and shapes. In robotics and optimization, manifolds serve as natural domains for configuration spaces and constraint-based problems. In data science, manifold learning and optimization on manifolds use geometric insights to model structured data.

Historically, ideas trace to Gauss and Riemann, with rigorous development by Levi-Civita, Cartan, Weyl, and Nash. The field continues to evolve, intersecting pure theory and practical computation across disciplines.