Introduction to manifolds, what does manifold mean

A coordinate map of a manifold, a coordinate graph, or simply a graph is a bijection between a subset of the manifold and a simple space, so that both the map and its inverse maintain the desired structure. For topological manifolds, the simple space is some Euclidean space rn and we are interested in its topology. This structure is maintained by the homomorphism, i.e. a reversible continuous mapping in both directions.

Graphs are extremely important for calculations because they allow the calculations to be performed in a simple space and then pass the results back to the manifold.

For example, polar coordinates are a graph of r2 in addition to the negative x-axis and origin. The mapping mentioned in the previous section top is a graph of a circle. Most manifolds require more than one graph (only the simplest manifold uses only one graph).

A collection of specific graphs that overlay manifolds is called an atlas. Atlases are not unique, as all manifolds can be overwritten in many ways by a combination of different graphs.

An atlas that contains all the plots that are consistent with a given atlas is called a very large atlas. Unlike a normal atlas, a very large atlas is unique. While it may be useful in definition, this object is very abstract and is not usually used directly (e.g., in calculations).

Atlases can also be used to define additional structures on manifolds. The structure is first defined separately on each diagram. If all transform mappings are compatible with this structure, the structure can be transferred to the manifold.

This is the standard way differential manifolds are defined. If the transformation mapping of the atlas retains the rn natural differential structure for a topological manifold (that is, if they are differential homomorphism), the differential structure is propagated to the manifold and becomes a differential manifold.

Usually, the structure of a manifold depends on an atlas, but sometimes different atlases give the same structure. Such an atlas is called compatible.

To put it simply, it is the same embryo in a local finite space

Explanation of manifolds (1).It is said that all things are nourished by nature and move and change their shapes. Yi Qian:

The clouds are raining, and the products are shaped. Gao Heng notes: "A manifold is the movement of its form.

These two sentences say that the sky has clouds and rain, and all things are nourished by it, and they can move their bodies in the universe. "Song Su Shi's "Confession to Wang Wen": Although the brilliance is colorful, it is not separated from the scattered manifolds.

Qing Ye Tingxuan's "Blowing Net Record: Yuan's Ode to Fenglongshan": Shen Xin Sense, Three Spirits Synthesis, Product Flow Shape, Nongkou Jiagu. ” 2).

The shape of the movement and change of all things. Jin Guo Pu "Jiang Fu": Huan the manifold of the block, mixed all in one subject.

Song Wen Tianxiang "Song of Righteousness": Heaven and earth have righteousness, and miscellaneous manifolds. Qing Liu Dazhu "Gift to Zhang Confucianism":

The eyes are noted, the hands are given, and they are reasonable, and they are all over the manifold, although they are suitable for the pills of the bureaucrats, the pours of the wheels, and the bearers of the people, and they think that they are more than that. "Morphology Words decomposition Explanation of flow flow ú liquid moves: flowing water.

Perspire. Bleed. Shed tears.

process. Diarrhoea. Liquid.

Running water does not rot. Sweat like a pig. Go with the flow (with the undulation of the waves, drifting with the flowing water, metaphorically having no opinion, following the current).

Flowing like water: circulation. Circulate.

Roving bandits. Stray. Interpretation of the Displacement Form Shape í Entity:

Shape instrument (posture meter). Form. Appearance.

Describe. The human body. Alone.

Inseparable. Appearance: Shaped sail malfunctional.

Form. Morphology. Traces.

Terrain. Situation. Performance:

Shape all the pen and ink. Joy in shape. Control, comparison:

Dwarfed by. Condition, topography: Situation.

Explanation of manifolds (1).It is said that all things are nourished by nature and move and change their shapes. Yi Qian:

The clouds are raining, and the products are shaped. Gao Heng notes: "A manifold is the movement of its form.

These two sentences say that the sky has clouds and rain, and all things are nourished by it, and they can move their bodies in the universe. "Song Su Shi's "Confession to Wang Wen": Although the brilliance is colorful, it is not separated from the scattered manifolds.

Qing Ye Tingxuan's "Blowing Net Record: Yuan's Ode to Fenglongshan": Shen Xin Sense, Three Spirits Synthesis, Product Flow Shape, Nongkou Jiagu. ” 2).

The shape of the movement and change of all things. Jin Guo Pu "Jiang Fu": Huan the manifold of the block, mixed all in one subject.

Song Wen Tianxiang "Song of Righteousness": Heaven and earth have righteousness, and miscellaneous manifolds. Qing Liu Dazhu "Gift to Zhang Confucianism":

The eyes are noted, the hands are given, and they are reasonable, and they are all over the manifold, although they are suitable for the pills of the bureaucrats, the pours of the wheels, and the bearers of the people, and they think that they are more than that. "Morphology Words decomposition Explanation of flow flow ú liquid moves: flowing water.

Perspire. Bleed. Shed tears.

process. Diarrhoea. Liquid.

Running water does not rot. Sweat like a pig. Go with the flow (with the undulation of the waves, drifting with the flowing water, metaphorically having no opinion, following the current).

Flowing like water: circulation. Circulate.

Roving bandits. Stray. Interpretation of the Displacement Form Shape í Entity:

Shape instrument (posture meter). Form. Appearance.

Describe. The human body. Alone.

Inseparable. Appearance: Shaped sail malfunctional.

Form. Morphology. Traces.

Terrain. Situation. Performance:

Shape all the pen and ink. Joy in shape. Control, comparison:

Dwarfed by. Condition, topography: Situation.

Entity. Let (x, or regalia) be the topological space, x1

x and x2x are any two independent points, and x1 exists respectively

Neighborhood n (x1

With n(x2, the slow greeting satisfies n(x1

n(x2= , then (x, ) is the hausdorff space. A is a subset of x, and if a is a closed set, then a is called an entity.

Manifold entities. x, ) is the Hausdorff space, a is the entity on x, and if the local topological dimension of any point inside A is n, then A is called an n-dimensional manifold entity; Conversely, the local topological dimension of any point inside the n-dimensional manifold entity is n. If there is a shirt which rock <>

If the local topological dimension d(x) is uncertain, then a is said to be a non-manifold entity. Group (a) is a manifold entity and group (b) is a non-manifold entity.

Manifold is a space with local Euclidean spatial properties, and is a generalization of the concepts of curves and surfaces in Euclidean space. Euclidean space is an example of the simplest manifold. A sphere such as the Earth's surface is a slightly more complex example.

A typical manifold can be formed by bending and gluing many straight sheets. Manifolds are used in mathematics to describe geometric shapes, and they provide a natural platform for studying the differentiability of shapes. Physically, the phase space of classical mechanics and the four-dimensional pseudo-Riemannian manifold that constructs the space-time model of general relativity are examples of manifolds.

Manifolds can also be defined in bitplace space. The torus is the two-pendulum space. In general, the topology of the geometry can be considered to be completely "soft", because all the deformations (symmorphism) will remain the topology; Analytic geometry is considered "hard" because the overall structure is fixed.

For example, in a polynomial, if you know the value of the interval, the value of the entire range of real numbers is fixed, so a local change will lead to a global change. A smooth manifold can be seen as a model in between: its infinitesimal structure is "hard", while the overall structure is "soft".

This may be the reason for the Chinese translation of the name "manifold" (the overall form can flow). The translation was introduced by Jiang Zehan, a well-known mathematician and mathematics educator. In this way, the hardness of the manifold allows it to accommodate differential structures, and its softness makes it useful as a mathematical and physical model for many local perturbations that require independence.

The easiest manifold to define is the topological manifold, which locally looks like some "ordinary" Euclidean space rn. Formally, a topological manifold is a topological space in which a local homomorphism is in a Euclidean space. This means that each point has a field that has a homomorphism (continuous bijection whose inverse is also continuous) that maps it to rn.

These isomorphisms are coordinate plots of manifolds.

Usually additional technical assumptions are added to this topological space to rule out pathological cases. The space can be requested as required to be Hausdorf's and second countable. This means that the line with two origins described below is not a topological manifold because it is not Hausdorf's.

The dimension of the manifold at a point is the dimension (the number n in the definition) of the Euclidean space graph to which the point is mapped. All points in a connected manifold have the same dimension. Some authors require that all graphs of a topological manifold be mapped to the same Euclidean space.

In this case, the topological space has a topological invariant, which is its dimension. Other authors allow topological manifolds to be non-intersecting and have different dimensions. Main article:

Differential manifolds. If the coordinate transformations between the local coordinate plots on the manifold are smooth, the direction, tangent space, and differentiable functions can be discussed on the manifold. In particular, "calculus" can be applied on differential manifolds. At this point we say that the manifold is endowed with a differential structure.

Manifolds with differential structures are called differential manifolds. If the coordinate transformation between any two local coordinates on a manifold is a "piecewise linear function", then we say that the manifold is given a piecewise linear structure. A topological manifold that is given a piecewise linear structure is called a piecewise linear manifold.

If there is a differential structure on the manifold, then the differential structure naturally induces a piecewise linear structure. So the differential manifold must be a piecewise linear manifold.

The presence of a piecewise linear structure is a slightly stronger condition than the presence of a simple partition; The category of piecewise linear manifolds is a category between the category of topological manifolds and the category of differential manifolds.

If we think of the topology of a geometry as completely soft, because all deformations (homomorphism) will keep the topology unchanged, and the analytic clusters as hard, because the overall structure is fixed (e.g. a 1-dimensional polynomial, if you know the value of the (0,1) interval, the value of the entire range of real numbers is fixed, and local perturbations will cause global changes), then we can think of a smooth manifold as a body in between, with an infinitesimal structure that is hard, And the overall structure is soft. This may be the reason for the Chinese translation of the manifold (the overall form can flow), which was introduced by the famous mathematician and mathematics educator Jiang Zehan. In this way, the hardness of the manifold allows it to accommodate differential structures, and its softness makes it a mathematical and physical model for many local perturbations that require independence.

Manifolds can be thought of as objects that look like Euclidean space or other relatively simple space in the near future. For example, people used to think that the earth was flat because we were small relative to it, which is an understandable illusion.

So, an ideal mathematical sphere in a small enough area also resembles a plane, which makes it a manifold. But a ball and a plane have a very different overall structure: if you walk in a fixed direction on a sphere, you end up back at the starting point, while on a plane, you can go all the way.

A surface is three-dimensional. However, manifolds can have arbitrary dimensions. Other examples are the circle of a line (one-dimensional) and all rotations in three-dimensional space (three-dimensional).

The example of the space made up of rotation shows that a manifold can be an abstract space. The technique of manifolds allows us to think about these objects independently, in the sense that we can have a sphere that is not dependent on any other space.

Local simplicity is a strong requirement. For example, we can't hang a line from a ball and call the whole a manifold; The area that contains the point where the line is glued to the ball is not simple—neither a line nor a surface—no matter how small the area is.

We sail the earth with flat maps collected in the atlas. Similarly, we can use a mathematical map (called a coordinate map) in a mathematical atlas to describe a manifold. It is often impossible to describe the entire manifold in one diagram because of the global structural differences between the manifold and the simple space used to build it.

When using multiple plots to cover manifolds, we must pay attention to the areas where they overlap, as these overlaps contain information about the overall structure.

There are many different kinds of manifolds. The simplest are topological manifolds, which locally look like Euclidean spaces. Other variants contain additional structures that are required for their use.

For example, a differential manifold supports not only topology, but also calculus. The idea of Riemann manifolds led to the mathematical basis of general relativity, which enabled people to describe space-time in terms of curvature.