Differentialgeomeetria

Differentialgeomeetria is a branch of mathematics that uses differential calculus, linear algebra, and tensor analysis to study geometric properties of smooth manifolds and the maps between them. It generalizes classical geometry by allowing the underlying spaces to be curved and by focusing on local infinitesimal data that determine global structure.

The subject centers on objects such as smooth manifolds, tangent spaces, vector and covector fields, differential forms, and more generally fiber bundles. A metric structure, most commonly a Riemannian metric, provides notions of length, angle, and distance on a manifold. A connection offers a way to differentiate vector fields along curves, leading to the notions of covariant derivative and parallel transport. The Levi-Civita connection is the canonical metric-compatible, torsion-free connection associated with a Riemannian metric.

Curvature is a central concept, quantifying how a space deviates from being flat. In Riemannian geometry, sectional curvature, Ricci curvature, and scalar curvature are used to describe curvature at different levels. Geodesics generalize straight lines to curved spaces as locally length-minimizing curves. The Gauss-Bonnet theorem links curvature to topology for surfaces, and higher-dimensional generalizations arise in Chern-Weil theory and related frameworks.

Differentialgeomeetria encompasses several major areas, including Riemannian geometry, symplectic geometry, complex differential geometry, and contact geometry. It relies on tools such as tensor calculus, exterior calculus, and the method of moving frames. Key results include Nash’s embedding theorem, which guarantees that abstract manifolds can be realized geometrically in Euclidean space, and a range of structure and rigidity theorems that connect curvature to global properties.

Applications of differentialgeomeetria appear in physics, notably general relativity, as well as computer graphics, vision, robotics, and the geometric analysis of data.