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 \(M\) be a Hausdorff and second countable topological space that is locally homeomorphic to Euclidean space \(\mathbb{R}^n\). We say that \(M\) is a manifold of dimension \(n\) if \(\forall p \in M, \exists U \subseteq M,\exists \phi: U \to \mathbb{R}^n\) such that \(\phi\) is a homeomorphism onto its image and \(\phi(U)\) is an open subset of \(\mathbb{R}^n\). Sometime we denote it as \(M^n\).

We denote \((U,\phi)\) as a coordinate chart, \(\phi\) as a coordinate map, and \(\phi^{-1}\) as a parameterization of \(U\).

Atlas and \(C^{k}\)-structure

An atlas on a topological manifold \(M\) is a collection of coordinate charts \(\mathcal{A} = \{(U_\alpha, \phi_\alpha)\}_{\alpha \in I}\) where \(\bigcup_{\alpha \in I} U_\alpha = M\).

Let \((U,\phi)\) and \((V, \psi)\) be two coordinate charts in the atlas \(\mathcal{A}\). The transition map from \((U, \phi)\) to \((V, \psi)\) is defined as \(\psi \circ \phi^{-1}: \phi(U \cap V) \to \psi(U \cap V)\).

We call the transition map \(C^{k}\) compatible if it is a \(C^{k}\) function. And we say that the atlas \(\mathcal{A}\) is a \(C^{k}\)-atlas if \(\forall (U, \phi), (V, \psi) \in \mathcal{A}, U\cap V \neq \emptyset\), the transition map \(\psi \circ \phi^{-1}\in C^{k}(\phi(U \cap V)\cap \psi(U \cap V))\).

We say \((U, \phi)\) are \(C^{k}\)-compatible with \(\mathcal{A}\) if it is \(C^{k}\) compatible with every chart in \(\mathcal{A}\).

Let \(\mathcal{A}\) be a \(C^{k}\)-atlas on a topological manifold \(M\). Maximal extension of \(\mathcal{A}\): \(\overline{\mathcal{A}} = \{(U, \phi) : (U, \phi) \text{ is } C^{k} \text{-compatible with } \mathcal{A}\}\). It is unique by definition and is a \(C^{k}\)-atlas on \(M\).

And it is not hard to prove that if \(\overline{\mathcal{A}} = \overline{\mathcal{B}}\) for another \(C^{k}\)-atlas \(\mathcal{B}\), then \(\mathcal{A} \cup \mathcal{B}\) is also a \(C^{k}\)-atlas on \(M\).

Actually, Let \(\mathcal{F}\) be the collection of all \(C^{k}\)-compatible coordinate charts on \(M\). Then \((\mathcal{F}, \subseteq)\) is a directed set. And \(\overline{\mathcal{A}}\) is the maximal element of \(\mathcal{F}\).

We call \(\overline{\mathcal{A}}\) a maximal \(C^{k}\)-atlas or a \(C^{k}\)-structure on \(M\). A manifold with a \(C^{k}\)-structure is called a \(C^{k}\)-manifold.

Smooth Manifolds

A smooth manifold is a \(C^{\infty}\)-manifold.

Coordinate Functions

Let \((U, \phi)\) be a coordinate chart on a \(n\)-manifold \(M\). The coordinate functions are the components of the coordinate map \(\phi: U \to \mathbb{R}^n\), denoted as \(\phi = (\phi^1, \phi^2, \ldots, \phi^n)\). We call the coordinate functions \(\phi^i: U \to \mathbb{R}\) the \(i\)-th coordinate function.

For \(p\in U\), we write \(\phi(p) = (x^1, x^2, \ldots, x^n)\), where \(x^i = \phi^i(p)\).

And we call \((x^1, x^2, \ldots, x^n)\) the local coordinates of \(p\) in the chart \((U, \phi)\).

Tangent Vectors and Tangent Spaces

Germs of functions

A germ of a function at a point \(p \in M\) is an equivalence class of functions defined on a neighborhood of \(p\). Two functions \(f\) and \(g\) are equivalent if they agree on some neighborhood of \(p\).

Formally, let \((f,U)\) and \((g,V)\) be two functions defined on neighborhoods \(U\) and \(V\) of \(p\). If there exists a neighborhood \(W \subseteq U \cap V\) such that \(f|_W = g|_W\), then we say that the germ of \(f\) at \(p\) is equal to the germ of \(g\) at \(p\), denoted as \((f,U) \sim (g,V)\).

A germ of a function at \(p\) is denoted as \([f]_p\), \([f]\) or \(f\) if the context is clear.

We denote \(C^{\infty}_p(M)\) or \(C^{\infty}_p\) as the set of germs of smooth functions at \(p \in M\). Formally, \(C^{\infty}_p(M) = \{ [f]_p : f \in C^{\infty}(U), U \text{ is a neighborhood of } p \}\).

If we write \(f \in C^{\infty}_p(M)\), it means that \([f]_p\in C^{\infty}_p(M)\).

Definition

There is three common definitions of tangent vectors on a manifold \(M\): 1. Tangent Vector as Derivation: A tangent vector at a point \(p \in M\) is a linear map \(v: C^{\infty}_p(M) \to \mathbb{R}\) that satisfies the Leibniz rule: \(v(fg) = v(f)g(p) + f(p)v(g)\) for all \(f, g \in C^{\infty}_p(M)\). 2. Tangent Vector as Equivalence Class of Curves: A tangent vector at a point \(p \in M\) is an equivalence class of curves \(\gamma: (-\epsilon, \epsilon) \to M\) such that \(\gamma(0) = p\). 3. Tangent Vector as Derivative of Coordinate Functions: A tangent vector at a point \(p \in M\) is a vector in the tangent space \(T_pM\) defined as the set of all derivations at \(p\) or equivalence classes of curves through \(p\).

The tangent space at a point \(p \in M\), denoted as \(T_pM\), is the set of all tangent vectors at \(p\). It is a vector space over \(\mathbb{R}\) of dimension \(n\), same as the dimension of the manifold \(M\).

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 \(\mathbb{R}\)-vector space structure.

Coordinate Expression

Let \((U,x)=(U,x^1,\ldots,x^n)\) be a coordinate chart containing \(p\).

Consider \(\left.\frac{\partial}{\partial x^i}\right|_p: C^{\infty}_p(M)\to \mathbb{R}\) act on \(f:U\to \mathbb R\) defined as \[\left.\frac{\partial}{\partial x^i}\right|_p([f])=\left.\frac{\partial(f\circ x^{-1})}{\partial x^i}\right |_{x(p)}\]

We know that \(\left \{\left.\frac{\partial}{\partial x^{1}}\right|_p,\ldots,\left.\frac{\partial}{\partial x^n}\right|_p\right \}\) forms a basis of \(T_p M\), $T_pM = M=n $.

By definition, we can prove the Kronecker delta property: \[ \left.\frac{\partial}{\partial x^i}\right|_p(x^j) = \frac{\partial x^j\circ x^{-1}}{\partial x^i} \bigg|_{x(p)} = \delta^j_i \] where \(\delta^i_j\) is the Kronecker delta, which is \(1\) if \(i=j\) and \(0\) otherwise.

So every \(X_p \in T_pM\) can be expressed as a linear combination of the basis vectors: \(X_p = \sum_{i=1}^n X_p^i \left.\frac{\partial}{\partial x^i}\right|_p\), where \(X_p^i =X_p(x^i)\in \mathbb{R}\).

The coordinate expression of a tangent vector \(X_p\) at \(p\) in the chart \((U,x)\) is given by the tuple \((a_1, a_2, \ldots, a_n)\), where \(a_i\) are the coefficients in the linear combination.

Fiber Bundles

Definition of Fiber Bundle

A fiber bundle is a structure \((E, M, \pi, F)\) consisting of:

• Total space: \(E\) (a manifold)
• Base space: \(M\) (a manifold)
• Bundle projection: \(\pi: E \to M\) (a smooth surjective map)
• Typical fiber: \(F\) (a manifold)

such that the local triviality condition holds: for each point \(p \in M\), there exists an open neighborhood \(U\) of \(p\) and a diffeomorphism \(\phi: \pi^{-1}(U) \to U \times F\) satisfying: \[ \text{pr}_1 \circ \phi = \pi|_{\pi^{-1}(U)} \] where \(\text{pr}_1: U \times F \to U\) is the projection onto the first factor.

The pair \((U, \phi)\) is called a local trivialization or bundle chart.

For each \(p \in M\), the fiber over \(p\) is defined as \(F_p = \pi^{-1}(p)\). The local triviality ensures that each fiber \(F_p\) is diffeomorphic to the typical fiber \(F\).

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 \((E, M, \pi, \mathbb{R}^k)\) where:

1. Each fiber \(F_p = \pi^{-1}(p)\) has the structure of a \(k\)-dimensional real vector space.
2. The local trivializations \(\phi: \pi^{-1}(U) \to U \times \mathbb{R}^k\) are linear on each fiber, meaning that for each \(p \in U\), the restriction \(\phi|_{F_p}: F_p \to \{p\} \times \mathbb{R}^k \cong \mathbb{R}^k\) is a vector space isomorphism.

The integer \(k\) is called the rank of the vector bundle.

Tangent Bundle

The tangent bundle of a manifold \(M\), denoted as \(TM\), is the vector bundle whose total space is the disjoint union of all tangent spaces: \[ TM = \bigcup_{p \in M} T_pM = \{(p, X_p) : p \in M, X_p \in T_pM\}\]

The bundle projection \(\pi: TM \to M\) is defined by \(\pi(p, X_p) = p\).

Local Trivialization: Let \((U, x = (x^1, \ldots, x^n))\) be a coordinate chart on \(M\). The tangent bundle can be locally trivialized over \(U\) by the map: \[ \Phi: \pi^{-1}(U) \to U \times \mathbb{R}^n \] \[ \Phi(p, X_p) = \left(p, (X_p^1, \ldots, X_p^n)\right) \] where \(X_p = \sum_{i=1}^n X_p^i \left.\frac{\partial}{\partial x^i}\right|_p\) is the coordinate representation of the tangent vector \(X_p\).

Manifold Structure: The tangent bundle \(TM\) is a smooth manifold of dimension \(2n\), where \(n\) is the dimension of \(M\). The coordinate charts on \(TM\) are given by \((U \times \mathbb{R}^n, \Phi^{-1})\) where \((U, x)\) ranges over all coordinate charts on \(M\).

Transition Maps: If \((U, x)\) and \((V, y)\) are two overlapping coordinate charts on \(M\), the transition map between the corresponding bundle charts is: \[ \Phi_V \circ \Phi_U^{-1}: (U \cap V) \times \mathbb{R}^n \to (U \cap V) \times \mathbb{R}^n \] \[ (p, (v^1, \ldots, v^n)) \mapsto \left(p, \left(\sum_{i=1}^n v^i \frac{\partial y^j}{\partial x^i}\bigg|_p\right)_{j=1}^n\right) \]

This map is smooth, confirming that \(TM\) has a smooth manifold structure.

Sections of Fiber Bundles

Definition of Sections

Let \(\pi: E \to M\) be a fiber bundle projection. A section of \(E\) is a map \(s: M \to E\) such that \(\pi \circ s = \text{id}_M\). In other words, for each point \(p \in M\), we have \(s(p) \in E_p\) (the fiber over \(p\)).

A section \(s\) is called smooth if \(s: M \to E\) is a smooth map between manifolds.

Local Expression of Smoothness

Let \((U, \phi)\) be a local trivialization of \(E\) over an open set \(U \subseteq M\), where \(\phi: \pi^{-1}(U) \to U \times F\) for some typical fiber \(F\). Any section \(s\) over \(U\) can be written as: \[ \phi \circ s|_U: U \to U \times F \] which has the form \(p \mapsto (p, f(p))\) for some function \(f: U \to F\).

Sometimes we just say \(f\) is a section of \(E\) over \(U\).

Space of Sections

We denote by \(\Gamma(E)\) or \(\Gamma(M, E)\) the space of all smooth sections of the fiber bundle \(E \to M\).

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

For Vector Bundles: When \(E \to M\) is a vector bundle with typical fiber \(\mathbb{R}^k\), the space \(\Gamma(E)\) has additional algebraic structures:

1. Vector space structure: For sections \(s_1, s_2 \in \Gamma(E)\) and scalars \(a, b \in \mathbb{R}\): \[ (as_1 + bs_2)(p) = as_1(p) + bs_2(p) \in E_p \]

2. \(C^\infty(M)\)-module structure: For a smooth function \(f \in C^\infty(M)\) and a section \(s \in \Gamma(E)\): \[ (fs)(p) = f(p) \cdot s(p) \in E_p \]

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

Differential

Let \(F: M^m \to N^n\) be a smooth map between manifolds \(M\) and \(N\). The differential of \(F\) at a point \(p \in M\), denoted as \(dF_p: T_pM \to T_{F(p)}N\), is a linear map between tangent spaces induced by the pushforward of \(F\).

It is defined as follows: for any tangent vector \(X_p \in T_pM\) and any smooth function \(f \in C^{\infty}_{F(p)}(N)\), \[ dF_p(X_p)f = X_p(f \circ F) \]

Local Coordinate Expression: Let \((U, x)\) and \((V, y)\) be coordinate charts around \(p\) and \(F(p)\) respectively, and let \(\tilde F = y \circ F \circ x^{-1}\) be the local representation of \(F\). Then: \[ \begin{align*} dF_p(X_p) &= \sum_{j=1}^n X_p(y^j\circ F) \frac{\partial}{\partial y^j} \bigg|_{F(p)}\\ &= \sum_{j=1}^n \sum_{i=1}^m X_p^i \frac{\partial \tilde F^j}{\partial x^i} \bigg|_{x(p)} \frac{\partial}{\partial y^j} \bigg|_{F(p)}\\ &= \sum_{j=1}^n \left(\sum_{i=1}^m X_p^i \frac{\partial \tilde F^j}{\partial x^i} \bigg|_{x(p)}\right) \frac{\partial}{\partial y^j} \bigg|_{F(p)} \end{align*} \]

In matrix form, if \(J_F(p) = \left(\frac{\partial \tilde F^j}{\partial x^i} \bigg|_{x(p)}\right)\) is the Jacobian matrix of \(F\) at \(p\), then: \[ dF_p(X_p) = J_F(p) X_p \] where \(X_p\) is considered as a column vector in local coordinates.

Vector Fields

A vector field on a manifold \(M\) is a smooth section of the tangent bundle \(TM\). That is, a vector field \(X\) is a smooth map \(X: M \to TM\) such that \(\pi \circ X = \text{id}_M\), where \(\pi: TM \to M\) is the bundle projection.

Equivalently, a vector field assigns to each point \(p \in M\) a tangent vector \(X_p \in T_pM\) in a smooth manner.

Local Coordinate Expression:

The partial derivative operator \(\frac{\partial}{\partial x^i}\) can be viewed as a vector field on \(M\) in the coordinate chart \((U, x)\), where it acts on smooth functions \(f \in C^\infty(U)\) by: \[ \frac{\partial f}{\partial x^i} = \left.\frac{\partial(f\circ x^{-1})}{\partial x^i}\right|_{x(p)} \]

where \(x^i\) is the \(i\)-th coordinate function in the chart \((U, x)\).

In a coordinate chart \((U, x = (x^1, \ldots, x^n))\), a vector field \(X\) can be expressed as: \[ X = \sum_{i=1}^n X^i \frac{\partial}{\partial x^i} \] where \(X^i: U \to \mathbb{R}\) are smooth functions called the components of the vector field \(X\) with respect to the coordinate chart \((U, x)\).

The smoothness of the vector field \(X\) is equivalent to the smoothness of all its component functions \(X^i\).

Space of Vector Fields: We denote by \(\mathfrak{X}(M)\) or \(\Gamma^{\infty}(TM)\) the space of all smooth vector fields on \(M\). This space is both a vector space over \(\mathbb{R}\) and a module over the ring \(C^\infty(M)\) of smooth functions on \(M\).

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 \(M\) is a smooth function \(\lambda: M \to \mathbb{R}\) 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 \(\mathbb{R}^n\). Consider the function \(\psi: \mathbb{R} \to \mathbb{R}\) defined by: \[ \psi(t) = \begin{cases} e^{-1/t} & \text{if } t > 0 \\ 0 & \text{if } t \le 0 \end{cases} \] This function is famously \(C^\infty\) on all of \(\mathbb{R}\), including at \(t=0\) where all its derivatives are zero. Using this, we can construct a smooth function on \(\mathbb{R}^n\) that is positive on an open ball and zero elsewhere. For example, the function \(\Psi: \mathbb{R}^n \to \mathbb{R}\) defined by \[ \Psi(x) = \begin{cases} e^{-1/(1-\|x\|^2)} & \text{if } \|x\| < 1 \\ 0 & \text{if } \|x\| \ge 1 \end{cases} \] is a smooth function that is positive on the open unit ball \(B(0,1)\) and has its support contained in the closed unit ball \(\overline{B(0,1)}\).

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 \(M\) be a smooth manifold, \(K \subset M\) be a compact set, and \(U \subset M\) be an open set containing \(K\). Then there exists a smooth function \(\lambda: M \to [0, 1]\) such that: 1. \(\lambda(p) = 1\) for all \(p \in K\). 2. \(\text{supp}(\lambda) \subset U\).

Here, the support of a function \(f: M \to \mathbb{R}\), denoted \(\text{supp}(f)\), is the closure of the set of points where \(f\) is non-zero: \[\text{supp}(f) = \overline{\{p \in M \mid f(p) \neq 0\}}\] A function with compact support is a function whose support is a compact set.

Definition of a Partition of Unity

Let \(M\) be a smooth manifold and let \(\mathcal{U} = \{U_\alpha\}_{\alpha \in A}\) be an open cover of \(M\). A family of smooth functions \(\{\rho_\alpha: M \to [0,1]\}_{\alpha \in A}\) indexed by the same set \(A\) is called a partition of unity subordinate to the cover \(\mathcal{U}\) if it satisfies the following conditions:

1. Subordination: For each \(\alpha \in A\), the support of \(\rho_\alpha\) is contained in \(U_\alpha\). That is, \(\text{supp}(\rho_\alpha) \subset U_\alpha\).
2. Local Finiteness: The cover of supports \(\{\text{supp}(\rho_\alpha)\}_{\alpha \in A}\) is locally finite. This means that for every point \(p \in M\), there exists a neighborhood \(V\) of \(p\) that intersects only a finite number of the sets \(\text{supp}(\rho_\alpha)\).
3. Sum to Unity: For every point \(p \in M\), the sum of the function values at that point is 1. \[\sum_{\alpha \in A} \rho_\alpha(p) = 1\] (The local finiteness condition ensures that for any \(p\), this is a finite sum in a neighborhood of \(p\), 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 \(M\) be a smooth manifold (which is Hausdorff and second-countable by our definition). For any open cover \(\mathcal{U}\) of \(M\), there exists a smooth partition of unity subordinate to \(\mathcal{U}\).

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 \(M\) is second-countable and locally compact, we can find a countable, locally finite open refinement \(\mathcal{V} = \{V_j\}_{j=1}^\infty\) of the original cover \(\mathcal{U}\) such that each \(\overline{V_j}\) is compact and for each \(j\), there is some \(\alpha_j\) with \(\overline{V_j} \subset U_{\alpha_j}\).
2. Find another refinement: We can construct another open cover \(\mathcal{W} = \{W_j\}_{j=1}^\infty\) such that for each \(j\), \(\overline{W_j}\) is compact and \(\overline{W_j} \subset V_j\).
3. Construct bump functions: For each \(j\), since \(\overline{W_j}\) is a compact set contained in the open set \(V_j\), we can use the bump function existence theorem to find a smooth function \(\psi_j: M \to [0,1]\) such that \(\psi_j \equiv 1\) on \(\overline{W_j}\) and \(\text{supp}(\psi_j) \subset V_j\).
4. Sum the bump functions: Define a function \(\Psi: M \to \mathbb{R}\) by \(\Psi(p) = \sum_{j=1}^\infty \psi_j(p)\). Since the cover \(\{V_j\}\) (and thus the supports of the \(\psi_j\)) is locally finite, this sum is finite in a neighborhood of any point, so \(\Psi\) is a smooth function. Since \(\{W_j\}\) is a cover, for any \(p \in M\), \(p \in W_j\) for some \(j\), which means \(\psi_j(p) = 1\). Therefore, \(\Psi(p) > 0\) for all \(p \in M\).
5. Normalize: For each \(j\), define \(\rho_j(p) = \frac{\psi_j(p)}{\Psi(p)}\). This family \(\{\rho_j\}_{j=1}^\infty\) is a partition of unity subordinate to the cover \(\mathcal{V}\), and therefore also subordinate to the original cover \(\mathcal{U}\).

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 \(f\) on a manifold \(M\), one can use a partition of unity \(\{\rho_\alpha\}\) subordinate to a cover of coordinate charts \(\{U_\alpha\}\). The integral is then defined as a sum of integrals over Euclidean space: \[\int_M f \, dV = \sum_\alpha \int_{U_\alpha} (\rho_\alpha f) \, dV = \sum_\alpha \int_{\phi_\alpha(U_\alpha)} (\rho_\alpha f) \circ \phi_\alpha^{-1} \, dx^1 \cdots dx^n\] The partition of unity ensures that each piece \((\rho_\alpha f)\) is compactly supported within a single chart, making the integral well-defined, and that the sum captures the “whole” of \(f\).

• Existence of Riemannian Metrics: Any smooth manifold admits a Riemannian metric. To prove this, one can define a Euclidean metric on each coordinate chart \((U_\alpha, \phi_\alpha)\). Using a partition of unity \(\{\rho_\alpha\}\) subordinate to the chart cover, one can patch these local metrics together into a global metric \(g\) via a weighted sum: \[g = \sum_\alpha \rho_\alpha g_\alpha\] where \(g_\alpha\) is the pullback of the Euclidean metric from \(\mathbb{R}^n\) to \(U_\alpha\). 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 \(\mathbb R^n_+ = \{(x^1, \ldots, x^n) \in \mathbb R^n : x^n \geq 0\}\) as the half-space in \(\mathbb R^n\).

A manifold with boundary is a topological space \(M\) that is Hausdorff, second countable and \(\forall p\in M\), there exists a neighborhood \(U\) of \(p\) that is homeomorphic to an open subset of \(\mathbb R^n_+\).

or equivalently, a Hausdorff, second countable topological space \(M\) is a manifold with boundary if it is locally homeomorphic to \(\mathbb R^n_+\) or \(\mathbb R^n\).

Let \(M\) be a manifold with boundary. The boundary of \(M\), denoted as \(\partial M\), is defined as the set of points in \(M\) that is not locally homeomorphic to \(\mathbb R^n\) i.e., \[ \partial M = \{ p \in M : \text{there is no neighborhood } U \cong \mathbb R^n \text{ around } p \} \] and \(\text{int}(M) = M \setminus \partial M\) is the interior of \(M\).

In other words, \(p\in \partial M\) if and only if there exists a local coordinate chart \((U,\phi)\) around \(p\) such that \(\phi(U) \subseteq \mathbb R^n_+\) and \(\phi(p) \in \partial \mathbb R^n_+= \{(x^1, \ldots, x^{n-1}, 0) : x^1, \ldots, x^{n-1} \in \mathbb R\}\).

We may denote \((M,\partial M)\) for a manifold with boundary and \(M\) can be called as manifold without boundary. Sometime \(M\) can be either situation depending on the context.

Properties: - \(\partial M\) is a manifold without boundary with dimension \(n-1\).

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

• For the tangent space of \((M,\partial M)\) remains a full space of \(\mathbb R^n\) at any point \(p\in M\).

Transversality and Intersection Theory