Manifold

In mathematics, a manifold is a special kind of space that looks flat and simple when you zoom in close to any point. Imagine taking a tiny piece of a surface—it might look like a flat piece of paper, even if the whole surface is curved like a ball or a donut. This idea helps mathematicians describe very complex shapes and spaces using simpler, flat pieces.

One-dimensional manifolds include simple shapes like lines and circles, but not more complicated curves that cross themselves. Two-dimensional manifolds, often called surfaces, include familiar shapes such as the plane, the sphere, the torus, the Klein bottle, and the real projective plane.

Manifolds are important in many areas of geometry and mathematical physics. They help describe complicated structures by breaking them into simpler parts. Manifolds appear naturally when solving groups of systems of equations or when looking at the graphs of functions. They are also used in computer graphics, for example, to turn medical images like CT scans into useful pictures with coordinates.

Manifolds can have extra rules added to them. For example, differentiable manifolds let us use calculus to study changes and motion. A Riemannian metric on a manifold lets us measure distances and angles. Other types of manifolds help describe the space and time used in theories like general relativity. Studying manifolds usually needs knowledge of calculus and topology.

Motivating examples

Circle

After a straight line, a circle is one of the simplest examples of a special kind of space called a manifold. Think of a circle as a curved line. If you take a small piece of a circle, it looks almost like a straight line because we are only looking at its shape, not how it is bent.

For example, look at just the top half of a circle, where the points are above the middle line. Each point on this top half can be described simply by its side-to-side position. This helps us understand how we can describe different parts of the circle using simple rules.

Sphere

A sphere, like the shape of a ball, is another example of a surface. To describe a sphere, we can use special rules for different parts of it. For example, we can split a sphere into top and bottom halves and describe each half separately.

Manifolds can be made of many pieces and do not have to be closed shapes like a ball. They can also be open, like a straight line that never ends. Some other examples of manifolds include curves like a parabola or a hyperbola.

However, some shapes are not manifolds. For example, two circles touching at one point, making a figure-8, cannot be described properly because at the touching point, the shape does not look like a simple line or curve.

Definition

Further information: Categories of manifolds

A manifold is a special kind of space that looks like regular flat space if you look at small parts of it. Think of it like a ball that looks almost flat when you zoom in very close.

Manifolds can be built using pieces that match up with flat space and fit together nicely. Some of these spaces can have extra rules that let us do more math with them smoothly.

Charts, atlases, and transition maps

Main article: Atlas (topology)

Manifold with boundary

See also: Topological manifold § Manifolds with boundary

A manifold with boundary is a special kind of shape that has an edge. For example, a flat circle with its inside (called a disk) is a two-dimensional shape with a circle as its edge. In three dimensions, a ball (a sphere with its inside) is a three-dimensional shape with a sphere as its edge.

In more technical terms, a manifold with boundary has both inside points and edge points. Each inside point has a nearby area that looks like a flat space, while edge points have areas that look like half of that flat space.

History

Further information: History of manifolds and varieties

The study of manifolds brings together many important parts of mathematics. It expands ideas about curves and surfaces, and connects to linear algebra and topology.

Early development

Before today’s idea of a manifold, there were many important discoveries.

Non-Euclidean geometry looks at spaces where Euclid’s rules don’t work. Saccheri studied these in 1733, but only to show they were wrong. Later, Gauss, Bolyai, and Lobachevsky found these spaces were real. They led to hyperbolic geometry and elliptic geometry. Today, these are types of manifolds with special curves.

Carl Friedrich Gauss was perhaps the first to think of spaces as objects on their own. His work showed how to measure the curve of a surface without looking at the space around it. This surface would now be called a manifold. The study of manifolds now looks only at these inner qualities, not the space around.

Another example is the Euler characteristic. Leonhard Euler found that for a shape in 3D space made of corners (V), edges (E), and faces (F), the formula V − E + F = 2 works. This stays true even if we draw the shape on a sphere. For a ring shape, called a torus, the answer is zero. This number helps us understand the shape’s topology.

Synthesis

Work by Niels Henrik Abel and Carl Gustav Jacobi in the 1800s led them to study special complex manifolds, now called Jacobians. Bernhard Riemann added more to this theory.

Manifolds also came from analytical mechanics, the study of how things move. The positions of moving objects can be thought of as points in a space, called phase space. This space is a high-dimensional manifold. For simple moves, it looks like normal space, but rules can make it more complex.

Riemann was the first to widely expand the idea of a surface to higher dimensions. The word manifold comes from his German term, Mannigfaltigkeit, meaning “many-valuedness.” He described sets with many values as a Mannigfaltigkeit. He separated these into continuous and discrete types. Riemann’s ideas grew into today’s formal idea of a manifold.

Poincaré's definition

In his important paper, Analysis Situs, Henri Poincaré gave an early definition of a manifold. He thought of it as a level set of a smooth function between Euclidean spaces.

Poincaré also suggested a new way to define manifolds using “chains of manifolds.” He looked at two manifolds defined by different functions. If they overlap, he said the coordinates must match smoothly in both directions. This was an early idea of a chart and transition map.

For example, the unit circle can be shown as the graph of a function near every point, except at (1, 0) and (−1, 0). Near those points, we can use a different function. Because the circle’s equation has no zero points except on the circle, every part of the circle is a submanifold of Euclidean space.

Hermann Weyl later gave an inner definition for manifolds, leading to today’s general idea of a topological space. In the 1930s, Hassler Whitney and others made the foundations clear. The Whitney embedding theorem showed that Poincaré’s idea matched the inner idea using charts.

Topology of manifolds: highlights

Two-dimensional manifolds, or 2D surfaces, were studied by Riemann as Riemann surfaces, and fully classified in the early 1900s by Poul Heegaard and Max Dehn. Poincaré began studying three-dimensional manifolds and asked a big question, now called the Poincaré conjecture. After almost 100 years, Grigori Perelman solved it. William Thurston extended this to all three-dimensional manifolds.

Four-dimensional manifolds became important in the 1980s through work by Michael Freedman and Simon Donaldson. They connected to physics ideas like Yang–Mills theory. Andrey Markov Jr. showed in 1960 that there is no way to classify four-dimensional manifolds by computer. Other mathematicians like René Thom, John Milnor, Stephen Smale, and Sergei Novikov did important work on higher-dimensional manifolds. Morse theory is a key tool used in studying the topology of manifolds.

Additional structure

Main listing: Categories of manifolds

Topological manifolds

Main article: topological manifold

The simplest kind of manifold is called a topological manifold. It looks locally like ordinary Euclidean space. This means every point has a nearby area that looks and acts like a part of regular space. These areas can be used to study the manifold, but they don't always let us measure distances or angles exactly the same way everywhere.

Differentiable manifolds

Main article: Differentiable manifold

For most uses, we need a special type of topological manifold called a differentiable manifold. This lets us use calculus to study the manifold. Each point has a tangent space, which is a flat space that helps us understand directions and changes at that point.

Riemannian manifolds

Main article: Riemannian manifold

To measure distances and angles on a manifold, it needs to be a Riemannian manifold. This means each tangent space has a way to measure distances between points, which changes smoothly from point to point. This lets us talk about lengths, angles, areas, and other shapes on the manifold.

Finsler manifolds

Main article: Finsler manifold

A Finsler manifold lets us define distances but not necessarily angles. Each tangent space has a way to measure sizes, which changes smoothly. This can be used to find the length of curves, but not always to measure angles between them.

Lie groups

Main article: Lie group

Lie groups are special kinds of manifolds that also work like groups in math. This means you can combine any two points in a smooth way to get another point. For example, rotating a circle is a Lie group because you can combine any two rotations to get another rotation.

Other types of manifolds

A complex manifold is studied in complex geometry. A one-dimensional complex manifold is called a Riemann surface.
A CR manifold is modeled on boundaries of areas in complex space.
'Infinite dimensional manifolds' include Banach manifolds and Fréchet manifolds.
A symplectic manifold is used in classical mechanics.
A combinatorial manifold is made from pieces that fit together.
A digital manifold is a special type found in digital space. See digital topology.

Classification and invariants

Further information: Classification of manifolds

In studying smooth closed shapes, mathematicians use special ways to tell them apart and understand their properties. For shapes that we can easily picture, like flat surfaces, we can use simple rules to decide if two shapes are the same. But for more complex shapes, it becomes very hard to know if they are the same or different.

Mathematicians use certain unchanging features, called invariants, to help tell shapes apart. These features stay the same no matter how you describe the shape. Even though it’s very hard to know if two complex shapes are exactly the same, scientists can often find ways to prove they are different using these invariants.

Important unchanging features include whether a shape is simply connected and whether it can be turned every which way without changing its basic nature. These ideas come from areas of math that study shapes and their properties.

Surfaces

Orientability

Main article: Orientable manifold

In mathematics, a surface is a special kind of space that looks flat when you zoom in on any part of it. For surfaces that are two-dimensional or more, an important question is whether they have a consistent "direction." Imagine looking at your hand—either it appears right-handed or left-handed. Some surfaces, like a sphere, can be drawn so that nearby parts match up in their handedness. These are called orientable surfaces. But other surfaces cannot match up this way, which means they are not orientable.

Examples of non-orientable surfaces include the Möbius strip, the Klein bottle, and the real projective plane. The Möbius strip is made by taking a strip, twisting one end, and gluing it back together. This creates a surface with only one side. The Klein bottle is made by joining two Möbius strips together. In normal three-dimensional space, it must pass through itself. The real projective plane is created by merging pairs of opposite points on a sphere.

Genus and the Euler characteristic

For flat, two-dimensional surfaces, a key feature is called the genus, which counts how many "handles" the surface has. A sphere has no handles, a torus (like a donut) has one handle, and a double torus has two handles. These features help describe the shape of the surface. In higher dimensions, this idea is replaced by the Euler characteristic and other mathematical tools.

Maps of manifolds

Main article: Maps of manifolds

There are many types of maps, or ways to connect, different manifolds. Some of these maps keep special shapes and sizes, like how a map of the world fits on a flat piece of paper.

One simple kind of map is a function that gives a number for each point on a manifold. These are used to study the shape and properties of the manifold itself.

Generalizations of manifolds

The idea of a manifold can be expanded in many ways. One way is to think about spaces that are like infinite-dimensional versions of regular manifolds. These spaces can be modeled on structures like Hilbert spaces, Banach spaces, and Fréchet spaces.

Another generalization is called an orbifold, which allows for certain kinds of "special points" in the space. These special points occur where groups act on the space in specific ways.

Algebraic varieties and schemes are also related to manifolds. They are built using solutions to polynomial equations and can have singularities, but they share some properties with manifolds.

Stratified spaces are spaces that can be broken into pieces, each of which is a manifold, fitting together in specific ways. Examples include manifolds with boundaries or corners.

CW-complexes are spaces made by gluing disks of different sizes together. While they are not always manifolds, they are important in the study of algebraic topology.

Homology manifolds behave like manifolds from the perspective of homology theory, even if they are not manifolds themselves.

Differential spaces are another generalization, defined using families of real functions and specific algebraic properties.