## Topological Manifold Wikipedia

### Topological manifold - Wikipedia.

Formal definition. A topological space X is called locally Euclidean if there is a non-negative integer n such that every point in X has a neighborhood which is homeomorphic to real n-space R n.. A topological manifold is a locally Euclidean Hausdorff space.It is common to place additional requirements on topological manifolds. In particular, many authors define them to be ....

https://en.wikipedia.org/wiki/Topological_manifold.

### Differentiable manifold - Wikipedia.

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus.Any manifold can be described by a collection of charts ().One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of ....

https://en.wikipedia.org/wiki/Differentiable_manifold.

### 4-manifold - Wikipedia.

In mathematics, a 4-manifold is a 4-dimensional topological manifold.A smooth 4-manifold is a 4-manifold with a smooth structure.In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be ....

https://en.wikipedia.org/wiki/4-manifold.

### Geometric topology - Wikipedia.

A handle decomposition of an m-manifold M is a union = = where each is obtained from by the attaching of -handles.A handle decomposition is to a manifold what a CW-decomposition is to a topological space--in many regards the purpose of a handle decomposition is to have a language analogous to CW-complexes, but adapted to the world of smooth manifolds. ....

https://en.wikipedia.org/wiki/Geometric_topology.

### Bifurcation theory - Wikipedia.

Bifurcation theory is the mathematical study of changes in the qualitative or topological structure of a given family of curves, such as the integral curves of a family of vector fields, and the solutions of a family of differential equations.Most commonly applied to the mathematical study of dynamical systems, a bifurcation occurs when a small smooth change made to the ....

https://en.wikipedia.org/wiki/Bifurcation_theory.

### Affine space - Wikipedia.

An affine transformation or endomorphism of an affine space is an affine map from that space to itself. One important family of examples is the translations: given a vector , the translation map : that sends + for every in is an affine map. Another important family of examples are the linear maps centred at an origin: given a point and a linear map , one may define an affine map ,: by.

https://en.wikipedia.org/wiki/Affine_space.

### Calabi–Yau manifold - Wikipedia.

In algebraic geometry, a Calabi-Yau manifold, also known as a Calabi-Yau space, is a particular type of manifold which has properties, such as Ricci flatness, yielding applications in theoretical physics.Particularly in superstring theory, the extra dimensions of spacetime are sometimes conjectured to take the form of a 6-dimensional Calabi-Yau manifold, which led to the idea of ....

https://en.wikipedia.org/wiki/Calabi%E2%80%93Yau_manifold.

### Fractal - Wikipedia.

In mathematics, fractal is a term used to describe geometric shapes containing detailed structure at arbitrarily small scales, usually having a fractal dimension strictly exceeding the topological dimension.Many fractals appear similar at various scales, as illustrated in successive magnifications of the Mandelbrot set. This exhibition of similar patterns at increasingly smaller ....

https://en.wikipedia.org/wiki/Fractal.

### 3-manifold - Wikipedia.

Introduction Definition. A topological space X is a 3-manifold if it is a second-countable Hausdorff space and if every point in X has a neighbourhood that is homeomorphic to Euclidean 3-space.. Mathematical theory of 3-manifolds. The topological, piecewise-linear, and smooth categories are all equivalent in three dimensions, so little distinction is made in whether we are dealing with ....

https://en.wikipedia.org/wiki/3-manifold.

### Pseudo-Riemannian manifold - Wikipedia.

A Lorentzian manifold is an important special case of a pseudo-Riemannian manifold in which the signature of the metric is (1, (-1)(n-1 occurrences) or (equivalently, (-1, 1(n-1 occurrences)) see Sign convention). ... there are certain topological obstructions. Furthermore, ....

https://en.wikipedia.org/wiki/Pseudo-Riemannian_manifold.

### Klein bottle - Wikipedia.

In topology, a branch of mathematics, the Klein bottle (/ ' k l aI n /) is an example of a non-orientable surface; it is a two-dimensional manifold against which a system for determining a normal vector cannot be consistently defined. Informally, it is a one-sided surface which, if traveled upon, could be followed back to the point of origin while flipping the traveler upside ....

https://en.wikipedia.org/wiki/Klein_bottle.

### Closed timelike curve - Wikipedia.

In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime, that is "closed", returning to its starting point.This possibility was first discovered by Willem Jacob van Stockum in 1937 and later confirmed by Kurt Godel in 1949, who discovered a solution to the equations of general relativity (GR) allowing CTCs known as ....

https://en.wikipedia.org/wiki/Closed_timelike_curve.

### Real coordinate space - Wikipedia.

In mathematics, the real coordinate space of dimension n, denoted R n (/ ?:r ' e n / ar-EN) or , is the set of the n-tuples of real numbers, that is the set of all sequences of n real numbers. With component-wise addition and scalar multiplication, it is a real vector space, and its elements are called coordinate vectors.. The coordinates over any basis of the elements of a real vector ....

https://en.wikipedia.org/wiki/Real_coordinate_space.

### Complex - Wikipedia.

Complex commonly refers to: . Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe . Complex system, a system composed of many components which may interact with each other; Complex (psychology), a core pattern of emotions etc. in the personal unconscious organized around a common theme ....

https://en.wikipedia.org/wiki/Complex.

### Surface - Wikipedia.

A surface, as the term is most generally used, is the outermost or uppermost layer of a physical object or space. It is the portion or region of the object that can first be perceived by an observer using the senses of sight and touch, and is the portion with which other materials first interact.The surface of an object is more than "a mere geometric solid", but is "filled with, ....

https://en.wikipedia.org/wiki/Surface.

### Poincaré conjecture - Wikipedia.

Henri Poincare was working on the foundations of topology--what would later be called combinatorial topology and then algebraic topology.He was particularly interested in what topological properties characterized a sphere.. Poincare claimed in 1900 that homology, a tool he had devised based on prior work by Enrico Betti, was sufficient to tell if a 3-manifold was a 3 ....

https://en.wikipedia.org/wiki/Poincar%C3%A9_conjecture.