differentiaalgeometry

Differential geometry is a branch of mathematics that uses calculus to study geometric properties of spaces that are locally similar to Euclidean space. The central objects are smooth manifolds, which come equipped with additional structures such as a metric, a connection, or a submanifold. These tools allow one to define notions of distance, angle, curvature, and parallel transport without relying on a fixed ambient coordinate system.

Riemannian geometry, a major subfield, begins with a smoothly varying inner product on each tangent space, called a Riemannian metric. This metric enables the definition of lengths of curves, distances between points, and shortest paths called geodesics. The Levi-Civita connection is a natural choice of parallel transport compatible with the metric, and it determines curvature through the Riemann curvature tensor.

Curvature measures how a space deviates from flat Euclidean space. Besides the full Riemann tensor, one studies sectional, Ricci, and scalar curvature. Submanifolds are studied via second fundamental form and Gauss equations. The Gauss-Bonnet theorem links curvature to topology for compact surfaces, while the Nash embedding theorem asserts that abstract manifolds can be isometrically embedded into Euclidean space.

Techniques include exterior calculus with differential forms, Cartan's structure equations, and, for Lie groups, connections and Maurer–Cartan theory. Computational tools and graphical methods also appear in numerical differential geometry and computer graphics.

Historically, differential geometry emerged from Gauss and Riemann in the 19th century and was developed further by Levi-Civita, Cartan, and many others. It plays a central role in general relativity, where spacetime is modeled as a four-dimensional pseudo-Riemannian manifold, and informs physics, robotics, computer vision, and geometric data analysis.