想起以前意识到一个问题。比如对于参数方程 y=f（x）因为只有 x 一个“维”,所以空间是一维的

(f(t), g(t)) 也是一维的

然后 (f(x, y) ,g(x, y)) 是二维的

这个东西有啥背景吗（

• One-Dimensional Manifolds: The sources describe one-dimensional manifolds as curves that can be represented by a single parameter. For instance:
• The graph of y = sin(x) is one-dimensional because a single parameter, x, determines a point on the curve.[1]
• Space curves, often described parametrically by equations like (x, y, z) = (f(t), g(t), h(t)), are also one-dimensional as they rely on a single parameter, t.[1]

• Two-Dimensional Manifolds: Two-dimensional manifolds, or surfaces, require two independent parameters to define a point. Examples include:
• A plane can be described using Cartesian coordinates (x, y) or polar coordinates (r, θ).
• A sphere can be represented using latitude and longitude.
• A torus can be parametrized using two angles.
• A surface defined by equations like x(u,v), y(u,v), z(u,v) explicitly uses two parameters, u and v, to specify a point.

• Higher-Dimensional Manifolds: The concept extends to higher dimensions. The unit 3-sphere in ℝ4, defined by the equation x12 + x22 + x32 + x42 = 1, is a three-dimensional manifold because three coordinates are sufficient to determine a nearby point within a neighborhood of any given point.

• Circle (1-manifold): A point on a circle can be identified by its angle. However, angles are periodic, meaning that angles differing by multiples of 2π represent the same point. Consequently, the relationship between angles and points on a circle is only locally one-to-one.

• Sphere (2-manifold): Latitude and longitude can define a point on a sphere. However, ambiguities arise at the poles (where longitude is undefined) and along the prime meridian (where longitude jumps by 360 degrees). Therefore, the correspondence between latitude/longitude and points on a sphere is also local.

but how to extend this relation to N

• Source Example 3.52: Discusses collapsing a subset A of a topological space X to a point, denoting the resulting quotient space as X/A. Although it doesn't explicitly mention N-dimensional spaces, it shows how spaces can be transformed and how their properties change under specific operations. This concept is relevant to understanding how dimensions might be reduced or increased through various mathematical operations.

• Source: Explains that the n-dimensional real projective space, denoted as Pn, is a topological n-manifold. It demonstrates that Pn can be covered by open sets, each homeomorphic to ℝn, implying that locally, Pn behaves like ℝn.

• Source: Shows that the n-sphere, Sn, can be given a smooth structure using stereographic coordinates. This example provides a concrete visualization of an n-dimensional manifold where points are defined using n parameters (coordinates).

• An N-dimensional manifold requires N independent parameters to specify a point within a local neighborhood.

• This relationship might not be globally one-to-one due to potential ambiguities in the parameterization.

Explanation of a Diffeomorphism

• Definition: The sources discuss smooth manifolds extensively. Source defines a topological manifold as "a second countable Hausdorff space in which every point has a neighborhood homeomorphic to an open subset of the closed n-dimensional upper half space."

• Smooth Structures: Source introduces the concept of a smooth structure on a manifold, which allows for the definition of smooth functions and maps on the manifold.

• Coordinate Representations: Source explains that a map between smooth manifolds is considered smooth if its coordinate representation with respect to any smooth charts is smooth. In essence, when expressed in local coordinates, the map should be differentiable infinitely many times.

• Formal Definition: A diffeomorphism between two smooth manifolds M and N is a bijective map F: M → N such that:
• F is smooth.
• F has an inverse map F-1: N → M.
• F-1 is also smooth.

• Equivalence of Smooth Structures: Diffeomorphisms establish a notion of equivalence between smooth manifolds. If there exists a diffeomorphism between two manifolds, they are considered "diffeomorphic." This means they have the same smooth structure and are essentially indistinguishable from the perspective of differential geometry.

• Preservation of Structure: Diffeomorphisms preserve essential geometric properties, such as:
• Dimension: The dimensions of diffeomorphic manifolds must be equal.
• Smooth Functions and Maps: Diffeomorphisms preserve the smoothness of functions and maps. The composition of a smooth map with a diffeomorphism is still smooth.
• Tangent Spaces: Diffeomorphisms induce isomorphisms between tangent spaces, preserving the local linear structure of manifolds.

• Homeomorphisms vs. Diffeomorphisms: It's important to distinguish diffeomorphisms from homeomorphisms. A homeomorphism is a continuous bijection with a continuous inverse, but it doesn't necessarily preserve the smooth structure.

• Stronger Condition: A diffeomorphism is a stronger condition than a homeomorphism. Every diffeomorphism is a homeomorphism, but not every homeomorphism is a diffeomorphism.