Differential geometry (nonfiction)

A triangle immersed in a saddle-shape plane (a hyperbolic paraboloid), as well as two diverging ultraparallel lines.

Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra, and multilinear algebra to study problems in geometry. The theory of plane and space curves and surfaces in the three-dimensional Euclidean space formed the basis for development of differential geometry during the 18th century and the 19th century.

Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds.

Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations.

The differential geometry of surfaces captures many of the key ideas and techniques in differential geometry.

History

Differential geometry arose and developed as a result of and in connection to the mathematical analysis of curves and surfaces. Mathematical analysis of curves and surfaces had been developed to answer some of the nagging and unanswered questions that appeared in calculus, like the reasons for relationships between complex shapes and curves, series and analytic functions. These unanswered questions indicated greater, hidden relationships.

The general idea of natural equations for obtaining curves from local curvature appears to have been first considered by Leonhard Euler in 1736, and many examples with fairly simple behavior were studied in the 1800s.

When curves, surfaces enclosed by curves, and points on curves were found to be quantitatively, and generally, related by mathematical forms, the formal study of the nature of curves and surfaces became a field of study in its own right, with Gaspard Monge's paper in 1795, and especially, with Carl Friedrich Gauss's publication of his Disquisitiones Generales Circa Superficies Curvas in Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores in 1827.

Initially applied to the Euclidean space, further explorations led to non-Euclidean space, and metric and topological spaces.

Branches

Branches of differential geometry include:

• Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric. This is a concept of distance expressed by means of a smooth positive definite symmetric bilinear form defined on the tangent space at each point. Riemannian geometry generalizes Euclidean geometry to spaces that are not necessarily flat, although they still resemble the Euclidean space at each point infinitesimally, i.e. in the first order of approximation. Various concepts based on length, such as the arc length of curves, area of plane regions, and volume of solids all possess natural analogues in Riemannian geometry.
• Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite. A special case of this is a Lorentzian manifold, which is the mathematical basis of Einstein's general relativity theory of gravity.
• Finsler geometry has Finsler manifolds as the main object of study. This is a differential manifold with a Finsler metric, that is, a Banach norm defined on each tangent space. Riemannian manifolds are special cases of the more general Finsler manifolds.

Fiction cross-reference

• Crimes against mathematical constants
• Gnomon algorithm
• Gnomon Chronicles
• Mathematics

Nonfiction cross-reference

• Differential calculus (nonfiction)
• Differential equation (nonfiction)
• Differential geometry of curves (nonfiction)
• Differential geometry of surfaces (nonfiction)
• Differential topology (nonfiction)
• Euclidean space (nonfiction)
• Finsler manifold (nonfiction)
• Integral (nonfiction)
• Linear algebra (nonfiction)
• Mathematics (nonfiction)
• Pseudo-Riemannian manifold (nonfiction)
• Riemannian geometry (nonfiction)

External links:

• Differential geometry @ Wikipedia