Differentiable manifold

smooth manifoldsmoothdifferential manifoldsmooth manifoldsdifferentiabledifferentiable manifoldsmanifoldmanifoldsambient manifoldCalculus on Manifolds
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.wikipedia
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Topological manifold

coordinate charttopological2-manifold
In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure.
All manifolds are topological manifolds by definition, but many manifolds may be equipped with additional structure (e.g. differentiable manifolds are topological manifolds equipped with a differential structure).

Differential structure

differentiable structuresmooth structure
In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure.
In mathematics, an n-dimensional differential structure (or differentiable structure) on a set M makes M into an n-dimensional differential manifold, which is a topological manifold with some additional structure that allows for differential calculus on the manifold.

Tangent space

tangent planetangenttangent vector
A differential structure allows one to define the globally differentiable tangent space, differentiable functions, and differentiable tensor and vector fields.
In differential geometry, one can attach to every point x of a differentiable manifold a tangent space—a real vector space that intuitively contains the possible directions in which one can tangentially pass through x. The elements of the tangent space at x are called the tangent vectors at x. This is a generalization of the notion of a bound vector in a Euclidean space.

Differential geometry

differentialdifferential geometerdifferential geometry and topology
This leads to such mathematical machinery as the exterior calculus. The study of calculus on differentiable manifolds is known as differential geometry.
Since the late 19th century, differential geometry has grown into a field concerned more generally with the geometric structures on differentiable manifolds.

Vector field

vector fieldsvectorgradient flow
A differential structure allows one to define the globally differentiable tangent space, differentiable functions, and differentiable tensor and vector fields.
More generally, vector fields are defined on differentiable manifolds, which are spaces that look like Euclidean space on small scales, but may have more complicated structure on larger scales.

Sheaf (mathematics)

sheafsheavessheaf theory
The structure sheaf of M, denoted C k, is a sort of functor that defines, for each open set U ⊂ M, an algebra C k (U) of continuous functions U → R. A structure sheaf C k is said to give M the structure of a C k manifold of dimension n provided that, for any p ∈ M, there exists a neighborhood U of p and n functions x 1, ..., x n ∈ C k (U) such that the map f = (x 1, ..., x n ) : U → R n is a homeomorphism onto an open set in R n, and such that C k | U is the pullback of the sheaf of k-times continuously differentiable functions on R n.
First, geometric structures such as that of a differentiable manifold or a scheme can be expressed in terms of a sheaf of rings on the space.

Cotangent space

co''tangent vectorcotangentcotangent vector
The cotangent space at a point is the dual of the tangent space at that point, and the cotangent bundle is the collection of all cotangent spaces.
In differential geometry, one can attach to every point x of a smooth (or differentiable) manifold, \mathcal M, a vector space called the cotangent space at x.

Tangent bundle

Canonical vector fieldrelative tangent bundletangent vector bundle
The collection of tangent spaces at all points can in turn be made into a manifold, the tangent bundle, whose dimension is 2n.
In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M.

Cotangent bundle

co''tangent bundlecotangent line bundlestandard symplectic form
The cotangent space at a point is the dual of the tangent space at that point, and the cotangent bundle is the collection of all cotangent spaces.
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold.

Algebraic geometry

complex algebraic geometryalgebraiccomputational algebraic geometry
This approach is strongly influenced by the theory of schemes in algebraic geometry, but uses local rings of the germs of differentiable functions.
One key achievement of this abstract algebraic geometry is Grothendieck's scheme theory which allows one to use sheaf theory to study algebraic varieties in a way which is very similar to its use in the study of differential and analytic manifolds.

Differentiable curve

differential geometry of curvescurvature vectortangent vector
Suppose that γ(t) is a curve in M with γ(0) = p, which is differentiable in the sense that its composition with any chart is a differentiable curve in R m.
Most of the notions presented here have analogues for curves in Riemannian and pseudo-Riemannian manifolds.

Smooth structure

Smooth atlasdifferentiable structure
A smooth manifold is a topological manifold M

Synthetic differential geometry

Hence, it is a more primitive definition of the structure (see synthetic differential geometry).
The first is that most of the analytic data for describing the class of smooth manifolds can be encoded into certain fibre bundles on manifolds: namely bundles of jets (see also jet bundle).

Hassler Whitney

WhitneyWhitney, Hassler
The widely accepted general definition of a manifold in terms of an atlas is due to Hassler Whitney.
In a 1936 paper, Whitney gave a definition of a smooth manifold of class ''

Symplectic manifold

Lagrangian submanifoldsymplecticspecial Lagrangian submanifold
The total space of a cotangent bundle has the structure of a symplectic manifold.
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.

Coordinate system

coordinatescoordinateaxis
The works of physicists such as James Clerk Maxwell, and mathematicians Gregorio Ricci-Curbastro and Tullio Levi-Civita led to the development of tensor analysis and the notion of covariance, which identifies an intrinsic geometric property as one that is invariant with respect to coordinate transformations.
For example, a differentiable manifold is a manifold where the change of coordinates from one coordinate map to another is always a differentiable function.

Diffeomorphism

diffeomorphicdiffeomorphismsdiffeomorphism group
Define a k-th order frame to be the k-jet of a diffeomorphism from R n to M.
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds.

Functor

contravariant functorfunctorialfunctors
The structure sheaf of M, denoted C k, is a sort of functor that defines, for each open set U ⊂ M, an algebra C k (U) of continuous functions U → R. A structure sheaf C k is said to give M the structure of a C k manifold of dimension n provided that, for any p ∈ M, there exists a neighborhood U of p and n functions x 1, ..., x n ∈ C k (U) such that the map f = (x 1, ..., x n ) : U → R n is a homeomorphism onto an open set in R n, and such that C k | U is the pullback of the sheaf of k-times continuously differentiable functions on R n.
Tangent and cotangent bundles: The map which sends every differentiable manifold to its tangent bundle and every smooth map to its derivative is a covariant functor from the category of differentiable manifolds to the category of vector bundles.

Holonomic basis

coordinate basisholonomicholonomic coordinates
For a set of (non-singular) coordinates x k local to the point, the coordinate derivatives define a holonomic basis of the tangent space.
In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold

Directional derivative

normal derivativedirectionalderivative
There are various ways to define the derivative of a function on a differentiable manifold, the most fundamental of which is the directional derivative.
be a differentiable manifold and

Exterior derivative

exterior calculusexterior differentiationdifferentials
This leads to such mathematical machinery as the exterior calculus. The study of calculus on differentiable manifolds is known as differential geometry.
On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree.

Differentiable function

differentiablecontinuously differentiabledifferentiability
If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart.
If M is a differentiable manifold, a real or complex-valued function f on M is said to be differentiable at a point p if it is differentiable with respect to some (or any) coordinate chart defined around p.

Affine connection

connectionaffineaffine connections
In the branch of mathematics called differential geometry, an affine connection is a geometric object on a smooth manifold which connects nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space.

Atlas (topology)

atlascharttransition map
The widely accepted general definition of a manifold in terms of an atlas is due to Hassler Whitney. Any manifold can be described by a collection of charts, also known as an atlas.
Such a manifold is called differentiable.

Immersion (mathematics)

immersionimmersedimmersions
Functions of maximal rank at a point are called immersions and submersions:
In mathematics, an immersion is a differentiable function between differentiable manifolds whose derivative is everywhere injective.