Differentiable manifold — Safekipedia Adventurer

In mathematics, a differentiable manifold is a special kind of space that looks like a vector space up close. This lets us use calculus to study it. We can imagine the manifold as many small pieces. Each piece is described by a chart, like pages in an atlas. These charts help us map the space into parts where we already know how to do calculus.

A nondifferentiable atlas of charts for the globe. The results of calculus may not be compatible between charts if the atlas is not differentiable. In the center and right charts, the Tropic of Cancer is a smooth curve, whereas in the left chart it has a sharp corner. The notion of a differentiable manifold refines that of a manifold by requiring the functions that transform between charts to be differentiable.

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. These smooth changes are called transition maps.

Differentiable manifolds are very important in physics. They help form 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. It helps us understand many natural phenomena.

History

Main article: History of manifolds and varieties

The idea of a differentiable manifold started with mathematicians like Carl Friedrich Gauss and Bernhard Riemann. Riemann talked about manifolds in a lecture, showing how objects can change in new directions. His work helped start the field of differential geometry.

Later, more ideas from mathematicians and physicists helped grow this concept. 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." These 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. There should be no sudden jumps or breaks. This smooth fitting together is what makes the whole space a differentiable manifold.

Given a topological space M...
a C^k atlas	is a collection of charts	{φ_α : U_α → Rⁿ}_α∈A	such that {U_α}_α∈A covers M, and such that for all α and β in A, the transition map φ_α ∘ φ⁻¹ _β	is a C^k map
a smooth or C^∞ atlas		{φ_α : U_α → Rⁿ}_α∈A		is a smooth map
is an analytic or C^ω atlas		{φ_α : U_α → Rⁿ}_α∈A		is a real-analytic map
is a holomorphic atlas		{φ_α : U_α → Cⁿ}_α∈A		is a holomorphic map

Differentiable functions

A differentiable manifold is a special kind of space where we can use calculus, 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. It follows the rules of calculus in every chart. No matter which chart we pick, the idea of smoothness stays the same. This smoothness helps 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 shows all the directions you can move from that point. It has the same number of directions as the dimension of the manifold. By gathering these tangent spaces from every point, we make the tangent bundle, which is itself a manifold. The tangent bundle has twice the dimension of the original manifold.

Further information: cotangent bundle

The cotangent space at a point is the space of special functions that work on the tangent space. Collecting these spaces creates the cotangent bundle, which is also a manifold. Cotangent vectors, sometimes called covectors, show how functions change at each point.

Further information: tensor bundle

The tensor bundle joins the tangent and cotangent bundles. Each part is a tensor field, which can work on vector fields in many ways. Though not a regular manifold because of its infinite dimension, it still follows certain algebraic rules.

Further information: frame bundle

A frame is an ordered set of directions at a point. The frame bundle gathers all possible frames at every point of the manifold. It helps us understand how tensor fields act when we change our view.

Further information: jet bundle

Jet bundles build on the ideas of tangent and cotangent bundles to include more detailed information about curves on the manifold. They are useful for studying differential operators on manifolds.

Calculus on manifolds

Many ideas from multivariate calculus also work on differentiable manifolds. We can study how functions change using something called the "differential." It works like a regular derivative but is used on special spaces.

There are also special rules, like the implicit and inverse function theorems, that help us understand how functions work.

We also use tools from integral calculus, like Green's theorem, the divergence theorem, and Stokes' theorem. These tools help us connect information about shapes and their edges in a 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 to study change and motion on curved spaces.

All differentiable manifolds can be described using maps called "charts." These charts are like maps of small pieces of the manifold. They help us understand the shape of the manifold by comparing it to simpler spaces, like straight lines. Mathematicians group these charts together if they work well together. This makes 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 type of smooth manifold. It has a way to measure distances and angles. This is done using something called an inner product. It 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 special structure.

A symplectic manifold is another type of smooth manifold. It has a special kind of form on it. This form helps us understand how things move and change on the manifold. These manifolds always have an even number of directions.

Main article: symplectic manifold

Main article: Lie group

A Lie group is a special kind of smooth manifold. It also has a group structure. This means we can combine points on the manifold in a way that follows the rules of group theory. 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 rules for how pieces of space can be linked together smoothly.

Another way to think about differentiable manifolds is through a structure sheaf. This looks at the functions on the manifold, instead of just the shapes we use to map it out. This method lets us describe the properties of the manifold using these functions.

Generalizations

Sometimes, mathematicians need more than just smooth manifolds. So, they created new ideas to build on them. For example, diffeological spaces use a different way to describe charts. Frölicher spaces and orbifolds are other approaches.

There are special types of manifolds called Banach manifolds and Fréchet manifolds. These have infinite dimensions and help us study spaces of mappings.

In non-commutative geometry, mathematicians study the algebra of functions on a manifold. They can use this algebra to rebuild the manifold. By looking at these algebras in new ways, they create ideas similar to manifolds using non-commutative algebras. This is the basis of the field of noncommutative geometry.