In mathematics, particularly topology, one describes a manifold using an **atlas**. An atlas consists of individual *charts* that, roughly speaking, describe individual regions of the manifold. If the manifold is the surface of the Earth, then an atlas has its more common meaning. In general, the notion of atlas underlies the formal definition of a manifold and related structures such as vector bundles and other fibre bundles.

The definition of an atlas depends on the notion of a *chart*. A **chart** for a topological space *M* (also called a **coordinate chart**, **coordinate patch**, **coordinate map**, or **local frame**) is a homeomorphism from an open subset *U* of *M* to an open subset of a Euclidean space. The chart is traditionally recorded as the ordered pair .

An **atlas** for a topological space is an indexed family of charts on which covers (that is, ). If the codomain of each chart is the *n*-dimensional Euclidean space, then is said to be an *n*-dimensional manifold.

The plural of atlas is *atlases*, although some authors use *atlantes*.^{ [1] }^{ [2] }

An atlas on an -dimensional manifold is called an **adequate atlas** if the image of each chart is either or , is a locally finite open cover of , and , where is the open ball of radius 1 centered at the origin and is the closed half space. Every second-countable manifold admits an adequate atlas.^{ [3] } Moreover, if is an open covering of the second-countable manifold then there is an adequate atlas on such that is a refinement of .^{ [3] }

A transition map provides a way of comparing two charts of an atlas. To make this comparison, we consider the composition of one chart with the inverse of the other. This composition is not well-defined unless we restrict both charts to the intersection of their domains of definition. (For example, if we have a chart of Europe and a chart of Russia, then we can compare these two charts on their overlap, namely the European part of Russia.)

To be more precise, suppose that and are two charts for a manifold *M* such that is non-empty. The **transition map** is the map defined by

Note that since and are both homeomorphisms, the transition map is also a homeomorphism.

One often desires more structure on a manifold than simply the topological structure. For example, if one would like an unambiguous notion of differentiation of functions on a manifold, then it is necessary to construct an atlas whose transition functions are differentiable. Such a manifold is called differentiable. Given a differentiable manifold, one can unambiguously define the notion of tangent vectors and then directional derivatives.

If each transition function is a smooth map, then the atlas is called a smooth atlas, and the manifold itself is called smooth. Alternatively, one could require that the transition maps have only *k* continuous derivatives in which case the atlas is said to be .

Very generally, if each transition function belongs to a pseudogroup of homeomorphisms of Euclidean space, then the atlas is called a -atlas. If the transition maps between charts of an atlas preserve a local trivialization, then the atlas defines the structure of a fibre bundle.

**Distributions**, also known as **Schwartz distributions** or **generalized functions**, are objects that generalize the classical notion of functions in mathematical analysis. Distributions make it possible to differentiate functions whose derivatives do not exist in the classical sense. In particular, any locally integrable function has a distributional derivative. Distributions are widely used in the theory of partial differential equations, where it may be easier to establish the existence of distributional solutions than classical solutions, or appropriate classical solutions may not exist. Distributions are also important in physics and engineering where many problems naturally lead to differential equations whose solutions or initial conditions are distributions, such as the Dirac delta function.

**Noether's theorem** or **Noether's first theorem** states that every differentiable symmetry of the action of a physical system has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in 1915 and published in 1918, after a special case was proven by E. Cosserat and F. Cosserat in 1909. The action of a physical system is the integral over time of a Lagrangian function, from which the system's behavior can be determined by the principle of least action. This theorem only applies to continuous and smooth symmetries over physical space.

In differential geometry, the **tangent bundle** of a differentiable manifold is a manifold which assembles all the tangent vectors in *. As a set, it is given by the disjoint union of the tangent spaces of **. That is,*

In the mathematical fields of differential geometry and tensor calculus, **differential forms** are an approach to multivariable calculus that is independent of coordinates. Differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics.

In mathematics, and particularly topology, a **fiber bundle** is a space that is *locally* a product space, but *globally* may have a different topological structure. Specifically, the similarity between a space and a product space is defined using a continuous surjective map

In mathematics, a **foliation** is an equivalence relation on an *n*-manifold, the equivalence classes being connected, injectively immersed submanifolds, all of the same dimension *p*, modeled on the decomposition of the real coordinate space **R**^{n} into the cosets *x* + **R**^{p} of the standardly embedded subspace **R**^{p}. The equivalence classes are called the **leaves** of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear, differentiable, or analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class *C ^{r}* it is usually understood that

In mathematics, and especially differential geometry and gauge theory, a **connection** on a fiber bundle is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. The most common case is that of a **linear connection** on a vector bundle, for which the notion of parallel transport must be linear. A linear connection is equivalently specified by a *covariant derivative*, an operator that differentiates sections of the bundle along tangent directions in the base manifold, in such a way that parallel sections have derivative zero. Linear connections generalize, to arbitrary vector bundles, the Levi-Civita connection on the tangent bundle of a Riemannian manifold, which gives a standard way to differentiate vector fields. Nonlinear connections generalize this concept to bundles whose fibers are not necessarily linear.

Suppose that *φ* : *M* → *N* is a smooth map between smooth manifolds *M* and *N*. Then there is an associated linear map from the space of 1-forms on *N* to the space of 1-forms on *M*. This linear map is known as the **pullback**, and is frequently denoted by *φ*^{∗}. More generally, any covariant tensor field – in particular any differential form – on *N* may be pulled back to *M* using *φ*.

In mathematics, an *n*-dimensional **differential 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. If *M* is already a topological manifold, it is required that the new topology be identical to the existing one.

In mathematics, a **differentiable manifold** is a type of manifold that is locally similar enough to a linear space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an atlas. One may then apply ideas from calculus while working within the individual charts, since each chart lies within a linear space to which the usual rules of calculus apply. If the charts are suitably compatible, then computations done in one chart are valid in any other differentiable chart.

In differential geometry, in the category of differentiable manifolds, a **fibered manifold** is a surjective submersion

In mathematics, specifically in symplectic geometry, the **momentum map** is a tool associated with a Hamiltonian action of a Lie group on a symplectic manifold, used to construct conserved quantities for the action. The momentum map generalizes the classical notions of linear and angular momentum. It is an essential ingredient in various constructions of symplectic manifolds, including **symplectic** (**Marsden–Weinstein**) **quotients**, discussed below, and symplectic cuts and sums.

In mathematics, in particular in nonlinear analysis, a **Fréchet manifold** is a topological space modeled on a Fréchet space in much the same way as a manifold is modeled on a Euclidean space.

In mathematics, a **holomorphic vector bundle** is a complex vector bundle over a complex manifold X such that the total space E is a complex manifold and the projection map π : *E* → *X* is holomorphic. Fundamental examples are the holomorphic tangent bundle of a complex manifold, and its dual, the holomorphic cotangent bundle. A **holomorphic line bundle** is a rank one holomorphic vector bundle.

In many-body theory, the term **Green's function** is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation and annihilation operators.

In mathematics, the **Kodaira–Spencer map**, introduced by Kunihiko Kodaira and Donald C. Spencer, is a map associated to a deformation of a scheme or complex manifold *X*, taking a tangent space of a point of the deformation space to the first cohomology group of the sheaf of vector fields on *X*.

In mathematics, a **harmonic morphism** is a (smooth) map between Riemannian manifolds that pulls back real-valued harmonic functions on the codomain to harmonic functions on the domain. Harmonic morphisms form a special class of harmonic maps i.e. those that are horizontally (weakly) conformal.

Martin Hairer's theory of **regularity structures** provides a framework for studying a large class of subcritical parabolic stochastic partial differential equations arising from quantum field theory. The framework covers the Kardar–Parisi–Zhang equation, the equation and the parabolic Anderson model, all of which require renormalization in order to have a well-defined notion of solution.

In Category theory and related fields of mathematics, an **envelope** is a construction that generalizes the operations of "exterior completion", like completion of a locally convex space, or Stone–Čech compactification of a topological space. A dual construction is called refinement.

In mathematics, and especially differential geometry and mathematical physics, **gauge theory** is the general study of connections on vector bundles, principal bundles, and fibre bundles. Gauge theory in mathematics should not be confused with the closely related concept of a gauge theory in physics, which is a field theory which admits gauge symmetry. In mathematics *theory* means a mathematical theory, encapsulating the general study of a collection of concepts or phenomena, whereas in the physical sense a gauge theory is a physical model of some natural phenomenon.

- ↑ Jost, Jürgen (11 November 2013).
*Riemannian Geometry and Geometric Analysis*. Springer Science & Business Media. ISBN 9783662223857 . Retrieved 16 April 2018– via Google Books. - ↑ Giaquinta, Mariano; Hildebrandt, Stefan (9 March 2013).
*Calculus of Variations II*. Springer Science & Business Media. ISBN 9783662062012 . Retrieved 16 April 2018– via Google Books. - 1 2 Kosinski, Antoni (2007).
*Differential manifolds*. Mineola, N.Y: Dover Publications. ISBN 978-0-486-46244-8. OCLC 853621933.

- Lee, John M. (2006).
*Introduction to Smooth Manifolds*. Springer-Verlag. ISBN 978-0-387-95448-6. - Sepanski, Mark R. (2007).
*Compact Lie Groups*. Springer-Verlag. ISBN 978-0-387-30263-8. - Husemoller, D (1994),
*Fibre bundles*, Springer, Chapter 5 "Local coordinate description of fibre bundles".

- Atlas by Rowland, Todd

This page is based on this Wikipedia article

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.

Text is available under the CC BY-SA 4.0 license; additional terms may apply.

Images, videos and audio are available under their respective licenses.