SafekipediaIn mathematics, a differentiable manifold is a special kind of space that looks like a vector space up close, which lets us use calculus to study it. We can think of the manifold as being made up of many small pieces, each described by a chart, similar to pages in an atlas. These charts help us map the space into parts where we already know how to do calculus.
A differentiable manifold is a type of topological manifold with a special structure called a differential structure. This structure lets us smoothly change from one chart to another where they overlap, making sure our calculus rules still work everywhere. These smooth changes are called transition maps.
Differentiable manifolds are very important in physics. They form the basis for theories like classical mechanics, general relativity, and Yang–Mills theory. By using these manifolds, scientists can describe complex shapes and motions in space. The study of calculus on these spaces is known as differential geometry, and it helps us understand many natural phenomena.
History
Main article: History of manifolds and varieties
The idea of a differentiable manifold began with mathematicians like Carl Friedrich Gauss and Bernhard Riemann. Riemann first talked about manifolds in a lecture, explaining how objects can change in new directions. His work helped create the field of differential geometry.
Later, the ideas of mathematicians and physicists helped develop this concept further. These ideas were important for Albert Einstein's theory of general relativity. Today, we use these ideas to understand shapes and spaces in advanced mathematics.
Definition
A differentiable manifold is a special kind of space that looks like regular flat space when you zoom in closely. This lets mathematicians use calculus — the study of change and motion — on these spaces.
To build a differentiable manifold, we use "charts," which are like maps that show small parts of the space and connect them to flat space. When two of these maps overlap, they must fit together smoothly, meaning there are no sudden jumps or breaks. This smooth fitting together is what makes the whole space a differentiable manifold.
| Given a topological space M... | ||||
|---|---|---|---|---|
| a Ck atlas | is a collection of charts | {φα : Uα → Rn}α∈A | such that {Uα}α∈A covers M, and such that for all α and β in A, the transition map φα ∘ φ−1 β | is a Ck map |
| a smooth or C ∞ atlas | {φα : Uα → Rn}α∈A | is a smooth map | ||
| is an analytic or C ω atlas | {φα : Uα → Rn}α∈A | is a real-analytic map | ||
| is a holomorphic atlas | {φα : Uα → Cn}α∈A | is a holomorphic map | ||
Differentiable functions
A differentiable manifold is a special kind of space where we can use calculus, just like we do with regular numbers. We describe the manifold using "charts," which are like maps that show small parts of the manifold as if they were flat spaces. This lets us use normal calculus rules in these small areas.
When we talk about a function being "differentiable" on a manifold, it means the function changes smoothly, following the rules of calculus in every chart. No matter which chart we pick, the idea of smoothness stays the same. This smoothness lets us study how functions change and move on the manifold using tools from calculus.
Bundles
Further information: tangent bundle
The tangent space at a point in a manifold includes all possible directions you can move from that point. It has the same number of directions as the dimension of the manifold. By collecting these tangent spaces at every point, we create the tangent bundle, which itself is a manifold. The tangent bundle is where tangent vectors live, and it has twice the dimension of the original manifold.
Further information: cotangent bundle
The cotangent space at a point is the space of linear functions that act on the tangent space. Collecting these spaces gives the cotangent bundle, which is also a manifold. Cotangent vectors, sometimes called covectors, measure how functions change at each point.
Further information: tensor bundle
The tensor bundle combines the tangent and cotangent bundles. Each element is a tensor field, which can act on vector fields in multiple ways. Though not a traditional manifold due to its infinite dimension, it still follows algebraic rules.
Further information: frame bundle
A frame is an ordered set of directions at a point. The frame bundle collects all possible frames at every point of the manifold. It helps in understanding how tensor fields behave under changes of perspective.
Further information: jet bundle
Jet bundles extend the idea of tangent and cotangent bundles to include higher-order information about curves on the manifold. They are important for studying differential operators on manifolds.
Calculus on manifolds
Many ideas from multivariate calculus work on differentiable manifolds too. We can talk about how functions change using something called the "differential," which behaves like the regular derivative from calculus but works on these special spaces. There are also special rules, like the implicit and inverse function theorems, that help us understand how functions behave.
We also have tools from integral calculus, like Green's theorem, the divergence theorem, and Stokes' theorem. These help us connect information about shapes and their edges in a very useful way.
Topology of differentiable manifolds
A differentiable manifold is a special kind of space that looks like straight lines when you zoom in very close. This lets mathematicians use calculus, which helps them study change and motion, on these curved spaces.
All differentiable manifolds can be described using maps called "charts," which are like maps of small pieces of the manifold. These charts help us understand the shape of the manifold by comparing it to simpler spaces we already know, like straight lines. Even though there can be many ways to make these charts, mathematicians group them together if they work well together, making the study of these spaces easier and more organized.
Structures on smooth manifolds
Main article: Riemannian manifold
Further information: Pseudo-Riemannian manifold
A Riemannian manifold is a special kind of smooth manifold that has a way to measure distances and angles on it. This is done using something called an inner product, which helps us understand how vectors relate to each other at each point. Because of this, we can talk about lengths, volumes, and angles on the manifold. Not every smooth manifold can have this kind of structure.
A symplectic manifold is another type of smooth manifold that has a special kind of form on it. This form helps us understand how things move and change on the manifold. These manifolds are always even-dimensional, meaning they can only have an even number of directions.
Main article: symplectic manifold
Main article: Lie group
A Lie group is a special kind of smooth manifold that also has a group structure. This means that we can combine points on the manifold in a way that follows the rules of group theory, and these combinations are smooth operations. Lie groups are important because they describe continuous symmetries in many areas of mathematics and physics.
Alternative definitions
The idea of a differentiable manifold can also be understood using something called a pseudogroup. This helps us describe many different types of structures on manifolds using a single method. A pseudogroup includes a set of rules for how pieces of space can be linked together smoothly.
Another way to think about differentiable manifolds is through something called a structure sheaf. This approach looks at the functions defined on the manifold itself, rather than just the shapes or "charts" we use to map it out. This method lets us describe the properties of the manifold using these functions, making it easier to work with in advanced mathematics.
Generalizations
The study of smooth manifolds sometimes needs extra features, so mathematicians have created new ideas to expand on them. For example, diffeological spaces use a different way to describe charts, and Frölicher spaces and orbifolds offer other approaches.
There are also special types of manifolds called Banach manifolds and Fréchet manifolds, which have infinite dimensions. These are useful for studying spaces of mappings.
In non-commutative geometry, mathematicians look at the algebra of functions on a manifold. This algebra can be used to rebuild the manifold itself. By thinking of these algebras in new ways, they can create ideas similar to manifolds but using non-commutative algebras, which is the basis of the field of noncommutative geometry.
This article is a child-friendly adaptation of the Wikipedia article on Differentiable manifold, available under CC BY-SA 4.0.
Images from Wikimedia Commons. Tap any image to view credits and license.