Differential geometry

differentialdifferential geometerdifferential geometry and topologydifferential geometricglobal differential geometrygeometricdifferential geometersAnalysis on Manifoldscomplex differential geometrygeometer
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.wikipedia
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Gaspard Monge

MongeMonge, Gaspard
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 Monge's paper in 1795, and especially, with Gauss's publication of his article, titled 'Disquisitiones Generales Circa Superficies Curvas', in Commentationes Societatis Regiae Scientiarum Gottingesis Recentiores in 1827.
Gaspard Monge, Comte de Péluse (9 May 1746 – 28 July 1818 ) was a French mathematician, the inventor of descriptive geometry (the mathematical basis of technical drawing), and the father of differential geometry.

Differential topology

topologydifferentialtopology of manifolds
Differential geometry is closely related to differential topology and the geometric aspects of the theory of differential equations.
It is closely related to differential geometry and together they make up the geometric theory of differentiable manifolds.

Linear algebra

linearlinear algebraiclinear-algebraic
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 telegraph required an explanatory system, and the 1873 publication of A Treatise on Electricity and Magnetism instituted a field theory of forces and required differential geometry for expression.

Differentiable manifold

smooth manifoldsmoothdifferential manifold
Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds. Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric.
This leads to such mathematical machinery as the exterior calculus. The study of calculus on differentiable manifolds is known as differential geometry.

Differential calculus

differentialdifferentiationcalculus
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.
Derivatives and their generalizations appear in many fields of mathematics, such as complex analysis, functional analysis, differential geometry, measure theory, and abstract algebra.

Mathematics

mathematicalmathmathematician
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.
Perhaps the foremost mathematician of the 19th century was the German mathematician Carl Friedrich Gauss, who made numerous contributions to fields such as algebra, analysis, differential geometry, matrix theory, number theory, and statistics.

Differentiable curve

differential geometry of curvescurvature vectortangent vector
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.
Differential geometry takes another path: curves are represented in a parametrized form, and their geometric properties and various quantities associated with them, such as the curvature and the arc length, are expressed via derivatives and integrals using vector calculus.

Gaussian curvature

Gauss curvaturecurvatureLiebmann's theorem
However, the Theorema Egregium of Carl Friedrich Gauss showed that for surfaces, the existence of a local isometry imposes strong compatibility conditions on their metrics: the Gaussian curvatures at the corresponding points must be the same.
In differential geometry, the Gaussian curvature or Gauss curvature Κ of a surface at a point is the product of the principal curvatures,

Symmetric space

Riemannian symmetric spacesymmetric spaceslocally symmetric space
An important class of Riemannian manifolds is the Riemannian symmetric spaces, whose curvature is not necessarily constant.
In differential geometry, representation theory and harmonic analysis, a symmetric space is a pseudo-Riemannian manifold whose group of symmetries contains an inversion symmetry about every point.

Riemann curvature tensor

Riemann tensorcurvature tensorcurvature
In higher dimensions, the Riemann curvature tensor is an important pointwise invariant associated with a Riemannian manifold that measures how close it is to being flat.
In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common way used to express the curvature of Riemannian manifolds.

Metric tensor

metricmetricsround metric
Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite.
In the mathematical field of differential geometry, a metric tensor is a type of function which takes as input a pair of tangent vectors v and w at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar

Pseudo-Riemannian manifold

pseudo-Riemannianpseudo-Riemannian metricpseudo
Pseudo-Riemannian geometry generalizes Riemannian geometry to the case in which the metric tensor need not be positive-definite.
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate.

Volume

volumetriccapacityOrders of magnitude (volume)
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.
In differential geometry, volume is expressed by means of the volume form, and is an important global Riemannian invariant.

Area

surface areaArea (geometry)area formula
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.
In addition to its obvious importance in geometry and calculus, area is related to the definition of determinants in linear algebra, and is a basic property of surfaces in differential geometry.

Symplectic geometry

symplectic topologysymplecticsymplectic structure
Symplectic geometry is the study of symplectic manifolds.
Symplectic geometry is a branch of differential geometry and differential topology that studies symplectic manifolds; that is, differentiable manifolds equipped with a closed, nondegenerate 2-form.

Symplectic manifold

Lagrangian submanifoldsymplecticspecial Lagrangian submanifold
Symplectic geometry is the study of symplectic manifolds.
In differential geometry, a subject of mathematics, a symplectic manifold is a smooth manifold, M, equipped with a closed nondegenerate differential 2-form \omega, called the symplectic form.

Geometry

geometricgeometricalgeometries
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.
In differential geometry and calculus, the angles between plane curves or space curves or surfaces can be calculated using the derivative.

Tensor

tensorsorderclassical treatment of tensors
The notion of a directional derivative of a function from multivariable calculus is extended in Riemannian geometry to the notion of a covariant derivative of a tensor.
The concept enabled an alternative formulation of the intrinsic differential geometry of a manifold in the form of the Riemann curvature tensor.

Surface (topology)

surfaceclosed surfacesurfaces
In dimension 2, a symplectic manifold is just a surface endowed with an area form and a symplectomorphism is an area-preserving diffeomorphism.
Topological surfaces are sometimes equipped with additional information, such as a Riemannian metric or a complex structure, that connects them to other disciplines within mathematics, such as differential geometry and complex analysis.

Tangent bundle

Canonical vector fieldrelative tangent bundletangent vector bundle
A contact structure on a (2n + 1)-dimensional manifold M is given by a smooth hyperplane field H in the tangent bundle that is as far as possible from being associated with the level sets of a differentiable function on M (the technical term is "completely nonintegrable tangent hyperplane distribution").
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M.

Darboux's theorem

Darboux coordinatesDarboux theoremPfaff problem
By contrast with Riemannian geometry, where the curvature provides a local invariant of Riemannian manifolds, Darboux's theorem states that all symplectic manifolds are locally isomorphic.
Darboux's theorem is a theorem in the mathematical field of differential geometry and more specifically differential forms, partially generalizing the Frobenius integration theorem.

Covariant derivative

covariant differentiationtensor derivativecovariant differential
The notion of a directional derivative of a function from multivariable calculus is extended in Riemannian geometry to the notion of a covariant derivative of a tensor.
In the 1940s, practitioners of differential geometry began introducing other notions of covariant differentiation in general vector bundles which were, in contrast to the classical bundles of interest to geometers, not part of the tensor analysis of the manifold.

Riemannian manifold

Riemannian metricRiemannianRiemannian manifolds
Riemannian geometry studies Riemannian manifolds, smooth manifolds with a Riemannian metric. An almost Hermitian structure is given by an almost complex structure J, along with a Riemannian metric g, satisfying the compatibility condition
In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M, g) is a real, smooth manifold M equipped with an inner product g p on the tangent space T p M at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p ↦ g p (X is a smooth function.

Kähler manifold

Kähler metricKählerKähler form
where \nabla is the Levi-Civita connection of g. In this case, (J, g) is called a Kähler structure, and a Kähler manifold is a manifold endowed with a Kähler structure.
In mathematics and especially differential geometry, a Kähler manifold is a manifold with three mutually compatible structures: a complex structure, a Riemannian structure, and a symplectic structure.

Hermitian manifold

Hermitian metricalmost Hermitian manifoldHermitian structure
An almost Hermitian structure is given by an almost complex structure J, along with a Riemannian metric g, satisfying the compatibility condition
In mathematics, and more specifically in differential geometry, a Hermitian manifold is the complex analogue of a Riemannian manifold.