Manifold

Klein bottle

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Polar stereographic projections

Circle with overlapping manifold charts

Circle manifold chart from slope

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Conics and cubic

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Sphere with chart

Manifold is a fundamental concept in the fields of mathematics, particularly in topology, geometry, and differential geometry. It refers to a topological space that resembles Euclidean space near each point. More formally, a manifold is a space that locally looks like Euclidean space of a specific dimension. This dimension is called the manifold's dimension. Manifolds serve as the main objects of study in differential geometry and are used to model a wide variety of physical systems and phenomena in physics, including the shape of the universe in general relativity, and the phase spaces in classical mechanics.

Definition

A manifold \(M\) of dimension \(n\) (denoted as an \(n\)-manifold) is a topological space where every point \(p\) in \(M\) has a neighborhood that is homeomorphic to an open subset of \(n\)-dimensional Euclidean space \(\mathbb{R}^n\). This local Euclidean property allows for the application of calculus, making manifolds the natural setting for differential calculus on spaces.

Types of Manifolds

There are several types of manifolds, each with specific properties and applications:

• Topological Manifold: A manifold that lacks additional structure beyond its topological properties. It is the most general type of manifold.

• Smooth Manifold: A manifold equipped with a smooth structure, allowing for the definition of differentiable functions. Smooth manifolds are central to differential geometry.

• Riemannian Manifold: A smooth manifold endowed with a Riemannian metric, which allows for the measurement of angles, lengths, and volumes. Riemannian manifolds are fundamental in the study of geometry and general relativity.

• Complex Manifold: A manifold that locally resembles complex Euclidean space and is equipped with a structure that makes it resemble a complex version of a smooth manifold. These are important in complex geometry and some areas of theoretical physics.

• Symplectic Manifold: A smooth manifold equipped with a symplectic form, which is a closed, non-degenerate 2-form. Symplectic manifolds are the mathematical backbone of classical mechanics and quantum mechanics.

Applications

Manifolds are used in many areas of mathematics and physics. In physics, the concept of a manifold underpins theories such as general relativity, where the fabric of spacetime is modeled as a 4-dimensional manifold. In robotics and control theory, manifolds represent the configuration spaces of mechanical systems. In mathematical visualization, manifolds help in understanding high-dimensional data structures through low-dimensional representations.

See Also

• Topology
• Differential Geometry
• Euclidean Space
• Homeomorphism
• Riemannian Geometry