Differential Manifold

References

• John Lee, Introduction to Smooth Manifolds

• Loring Tu, An Introduction to Manifolds

• Victor Guillemin and Alan Pollack, Differential Topology

For Chinese reader, you can refer to the website http://staff.ustc.edu.cn/~wangzuoq/Courses/23F-Manifolds/#notes for Chinese course note in USTC.

Manifolds and Smooth maps

Topological Manifolds

Let be a Hausdorff and second countable topological space that is locally homeomorphic to Euclidean space . We say that is a manifold of dimension if such that is a homeomorphism onto its image and is an open subset of . Sometime we denote it as .

We denote as a coordinate chart, as a coordinate map, and as a parameterization of .

Atlas and -structure

An atlas on a topological manifold is a collection of coordinate charts where .

Let and be two coordinate charts in the atlas . The transition map from to is defined as .

We call the transition map compatible if it is a function. And we say that the atlas is a -atlas if , the transition map .

We say are -compatible with if it is compatible with every chart in .

Let be a -atlas on a topological manifold . Maximal extension of : . It is unique by definition and is a -atlas on .

And it is not hard to prove that if for another -atlas , then is also a -atlas on .

Actually, Let be the collection of all -compatible coordinate charts on . Then is a directed set. And is the maximal element of .

We call a maximal -atlas or a -structure on . A manifold with a -structure is called a -manifold.

Smooth Manifolds

A smooth manifold is a -manifold.

Coordinate Functions

Let be a coordinate chart on a -manifold . The coordinate functions are the components of the coordinate map , denoted as . We call the coordinate functions the -th coordinate function.

For , we write , where .

And we call the local coordinates of in the chart .

Tangent Vectors and Tangent Spaces

Germs of functions

A germ of a function at a point is an equivalence class of functions defined on a neighborhood of . Two functions and are equivalent if they agree on some neighborhood of .

Formally, let and be two functions defined on neighborhoods and of . If there exists a neighborhood such that , then we say that the germ of at is equal to the germ of at , denoted as .

A germ of a function at is denoted as , or if the context is clear.

We denote or as the set of germs of smooth functions at . Formally, .

If we write , it means that .

Definition

There is three common definitions of tangent vectors on a manifold : 1. Tangent Vector as Derivation: A tangent vector at a point is a linear map that satisfies the Leibniz rule: for all . 2. Tangent Vector as Equivalence Class of Curves: A tangent vector at a point is an equivalence class of curves such that . 3. Tangent Vector as Derivative of Coordinate Functions: A tangent vector at a point is a vector in the tangent space defined as the set of all derivations at or equivalence classes of curves through .

The tangent space at a point , denoted as , is the set of all tangent vectors at . It is a vector space over of dimension , same as the dimension of the manifold .

We shall mainly focus on the first definition of tangent vectors as derivations, which is the most general and abstract definition.

The tangent space is equipped with a natural -vector space structure.

Coordinate Expression

Let be a coordinate chart containing .

Consider act on defined as

We know that forms a basis of , .

By definition, we can prove the Kronecker delta property: where is the Kronecker delta, which is if and otherwise.

So every can be expressed as a linear combination of the basis vectors: , where .

The coordinate expression of a tangent vector at in the chart is given by the tuple , where are the coefficients in the linear combination.

Fiber Bundles

Definition of Fiber Bundle

A fiber bundle is a structure consisting of:

• Total space: (a manifold)
• Base space: (a manifold)
• Bundle projection: (a smooth surjective map)
• Typical fiber: (a manifold)

such that the local triviality condition holds: for each point , there exists an open neighborhood of and a diffeomorphism satisfying: where is the projection onto the first factor.

The pair is called a local trivialization or bundle chart.

For each , the fiber over is defined as . The local triviality ensures that each fiber is diffeomorphic to the typical fiber .

Intuitively, a fiber bundle describes a space that ‘locally looks like a product’ but may have global twisting, like the Möbius strip over a circle.

Vector Bundles

A vector bundle is a fiber bundle where:

1. Each fiber has the structure of a -dimensional real vector space.
2. The local trivializations are linear on each fiber, meaning that for each , the restriction is a vector space isomorphism.

The integer is called the rank of the vector bundle.

Tangent Bundle

The tangent bundle of a manifold , denoted as , is the vector bundle whose total space is the disjoint union of all tangent spaces:

The bundle projection is defined by .

Local Trivialization: Let be a coordinate chart on . The tangent bundle can be locally trivialized over by the map: where is the coordinate representation of the tangent vector .

Manifold Structure: The tangent bundle is a smooth manifold of dimension , where is the dimension of . The coordinate charts on are given by where ranges over all coordinate charts on .

Transition Maps: If and are two overlapping coordinate charts on , the transition map between the corresponding bundle charts is:

This map is smooth, confirming that has a smooth manifold structure.

Sections of Fiber Bundles

Definition of Sections

Let be a fiber bundle projection. A section of is a map such that . In other words, for each point , we have (the fiber over ).

A section is called smooth if is a smooth map between manifolds.

Local Expression of Smoothness

Let be a local trivialization of over an open set , where for some typical fiber . Any section over can be written as: which has the form for some function .

Sometimes we just say is a section of over .

Space of Sections

We denote by or the space of all smooth sections of the fiber bundle .

For General Fiber Bundles: The space has the structure of a set with pointwise operations (when they make sense on the typical fiber).

For Vector Bundles: When is a vector bundle with typical fiber , the space has additional algebraic structures:

1. Vector space structure: For sections and scalars :

2. -module structure: For a smooth function and a section :

These structures exist because each fiber is a vector space, allowing us to perform linear operations.

Differential

Let be a smooth map between manifolds and . The differential of at a point , denoted as , is a linear map between tangent spaces induced by the pushforward of .

It is defined as follows: for any tangent vector and any smooth function ,

Local Coordinate Expression: Let and be coordinate charts around and respectively, and let be the local representation of . Then:

In matrix form, if is the Jacobian matrix of at , then: where is considered as a column vector in local coordinates.

Vector Fields

A vector field on a manifold is a smooth section of the tangent bundle . That is, a vector field is a smooth map such that , where is the bundle projection.

Equivalently, a vector field assigns to each point a tangent vector in a smooth manner.

Local Coordinate Expression:

The partial derivative operator can be viewed as a vector field on in the coordinate chart , where it acts on smooth functions by:

where is the -th coordinate function in the chart .

In a coordinate chart , a vector field can be expressed as: where are smooth functions called the components of the vector field with respect to the coordinate chart .

The smoothness of the vector field is equivalent to the smoothness of all its component functions .

Space of Vector Fields: We denote by or the space of all smooth vector fields on . This space is both a vector space over and a module over the ring of smooth functions on .

Partition of Unity

Partitions of unity are one of the most powerful tools in the theory of smooth manifolds. They provide a way to smoothly “glue together” local constructions into global ones. For example, if we can define an object (like a function, a metric, or a differential form) on each chart of an atlas, a partition of unity allows us to combine these local objects into a single, globally defined smooth object on the entire manifold. The fundamental building blocks for partitions of unity are smooth bump functions.

Bump Functions

A bump function on a manifold is a smooth function that is equal to 1 on some specified compact set and is zero outside of a slightly larger open set containing it. The existence of such functions is a cornerstone of analysis on manifolds, distinguishing smooth manifolds from, for example, analytic manifolds where such functions do not exist.

First, let’s establish their existence on . Consider the function defined by: This function is famously on all of , including at where all its derivatives are zero. Using this, we can construct a smooth function on that is positive on an open ball and zero elsewhere. For example, the function defined by is a smooth function that is positive on the open unit ball and has its support contained in the closed unit ball .

By scaling and translating this function, we can create a bump function for any ball. More generally, we have the following crucial existence theorem for bump functions on manifolds:

Theorem (Existence of Bump Functions): Let be a smooth manifold, be a compact set, and be an open set containing . Then there exists a smooth function such that: 1. for all . 2. .

Here, the support of a function , denoted , is the closure of the set of points where is non-zero: A function with compact support is a function whose support is a compact set.

Definition of a Partition of Unity

Let be a smooth manifold and let be an open cover of . A family of smooth functions indexed by the same set is called a partition of unity subordinate to the cover if it satisfies the following conditions:

1. Subordination: For each , the support of is contained in . That is, .
2. Local Finiteness: The cover of supports is locally finite. This means that for every point , there exists a neighborhood of that intersects only a finite number of the sets .
3. Sum to Unity: For every point , the sum of the function values at that point is 1. (The local finiteness condition ensures that for any , this is a finite sum in a neighborhood of , and thus the total sum is a well-defined smooth function.)

Existence of Partitions of Unity

The main theorem guarantees that such partitions of unity always exist on manifolds that satisfy the standard topological assumptions.

Theorem (Existence of Partitions of Unity): Let be a smooth manifold (which is Hausdorff and second-countable by our definition). For any open cover of , there exists a smooth partition of unity subordinate to .

Sketch of Proof: The proof relies on the topological property of paracompactness, which is guaranteed for Hausdorff, second-countable manifolds.

1. Find a good refinement: Since is second-countable and locally compact, we can find a countable, locally finite open refinement of the original cover such that each is compact and for each , there is some with .
2. Find another refinement: We can construct another open cover such that for each , is compact and .
3. Construct bump functions: For each , since is a compact set contained in the open set , we can use the bump function existence theorem to find a smooth function such that on and .
4. Sum the bump functions: Define a function by . Since the cover (and thus the supports of the ) is locally finite, this sum is finite in a neighborhood of any point, so is a smooth function. Since is a cover, for any , for some , which means . Therefore, for all .
5. Normalize: For each , define . This family is a partition of unity subordinate to the cover , and therefore also subordinate to the original cover .

Applications of Partitions of Unity

Partitions of unity are essential for extending local properties to global ones. Here are two classic applications:

• Integration on Manifolds: To define the integral of a function on a manifold , one can use a partition of unity subordinate to a cover of coordinate charts . The integral is then defined as a sum of integrals over Euclidean space: The partition of unity ensures that each piece is compactly supported within a single chart, making the integral well-defined, and that the sum captures the “whole” of .

• Existence of Riemannian Metrics: Any smooth manifold admits a Riemannian metric. To prove this, one can define a Euclidean metric on each coordinate chart . Using a partition of unity subordinate to the chart cover, one can patch these local metrics together into a global metric via a weighted sum: where is the pullback of the Euclidean metric from to . The result is a smoothly varying inner product on each tangent space, i.e., a Riemannian metric.

Local Behavior of Smooth Maps (Submersion, Immersion, Embedding)

Homotopy

Sard’s theorem

Submanifold

Whitney Embedding Theorem

Tubular Neighborhood Theorem

Manifold with Boundary

First, we denote as the half-space in .

A manifold with boundary is a topological space that is Hausdorff, second countable and , there exists a neighborhood of that is homeomorphic to an open subset of .

or equivalently, a Hausdorff, second countable topological space is a manifold with boundary if it is locally homeomorphic to or .

Let be a manifold with boundary. The boundary of , denoted as , is defined as the set of points in that is not locally homeomorphic to i.e., and is the interior of .

In other words, if and only if there exists a local coordinate chart around such that and .

We may denote for a manifold with boundary and can be called as manifold without boundary. Sometime can be either situation depending on the context.

Properties: - is a manifold without boundary with dimension .

• The result of atlas on manifold with boundary is similar.

• For the tangent space of remains a full space of at any point .

Transversality and Intersection Theory