differentialgeometric

Differential geometry, sometimes written as differential-geometric, is the branch of mathematics that uses differential calculus, along with topology and algebra, to study geometric properties of smooth spaces. It provides a framework to analyze shapes, curves, and surfaces with notions of distance, angle, and curvature.

The central objects are differentiable manifolds; tangent spaces; vector fields; differential forms; and connections. A Riemannian metric defines lengths and angles; curvature measures deviance from flatness, with the Riemann curvature tensor as a primary invariant. Tools from linear algebra and analysis, such as tensors and exterior calculus, enable precise formulations of geometric phenomena.

Geodesics, the shortest paths between points, are defined via connections; the Levi-Civita connection is the unique metric-compatible one. Core results include the Gauss-Bonnet theorem, which links curvature to topological invariants, and various comparison theorems that relate local geometry to global structure. Intrinsic geometry studies properties that depend only on the manifold’s own structure, rather than any embedding into a higher-dimensional space.

Subfields and related areas include Riemannian geometry, pseudo-Riemannian geometry (notably Lorentzian geometry in relativity), complex differential geometry, symplectic geometry, and contact geometry. Differential-geometric methods underlie much of general relativity, gauge theory, geometric analysis, and parts of mathematical physics, as well as applications in computer graphics and robotics.

Historical roots trace to Gauss and Riemann, with later foundational contributions from Cartan, Weyl, and Ehresmann. Today, the field blends analysis, topology, and algebra to study the shape of space and its physical and mathematical implications.