Differential geometry

Differential geometry studies smooth spaces and geometric structures using differential calculus. It analyzes local quantities such as curvature and how they relate to global shape.

Smooth manifold: A space that locally resembles Euclidean space and has smoothly compatible coordinate charts. It is the underlying space on which differential geometry defines smooth structures.
Riemannian metric: A smoothly varying positive-definite inner product on the tangent spaces of a manifold. It defines lengths, angles, and distances on the spaces studied in Riemannian geometry.
General relativity: A theory of gravitation in which spacetime geometry is dynamic and shaped by matter and energy. Its field equations use curvature to connect spacetime geometry with physical sources.
Topology: The study of properties of spaces preserved under continuous deformation. Topology ignores many metric details that differential geometry measures through smooth structures.
Carl Friedrich Gauss: A German mathematician whose work established foundational results in number theory, analysis, and geometry. His Theorema Egregium showed that Gaussian curvature is intrinsic to a surface.
Tangent space: A vector space that collects the possible tangent directions at a point of a manifold. It lets calculus describe directions and derivatives at points on curved spaces.
Affine connection: A rule for differentiating vector fields along other vector fields on a manifold. It makes directional differentiation possible when tangent spaces at different points differ.
Gaussian curvature: The product of the principal curvatures of a surface at a point. It measures intrinsic surface curvature and appears in local and global surface results.
Algebraic geometry: The study of geometric spaces defined by polynomial equations using algebraic methods. It often builds geometry from algebraic equations rather than differential calculus.
Bernhard Riemann: A German mathematician who introduced broad ideas about manifolds and geometric spaces. His 1854 lecture developed the foundations of geometry on curved spaces.

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