Smooth Manifolds | Mathematics

An $n$-dimensional topological manifold $M$ is a topological space that is:

1. Hausdorff: distinct points have disjoint neighborhoods.
2. Second countable: the topology has a countable basis.
3. Locally Euclidean: every point $p \in M$ has a neighborhood $U$ homeomorphic to an open subset of $\mathbb{R}^n$.

A homeomorphism $\varphi : U \to V \subseteq \mathbb{R}^n$ is called a coordinate chart (or just chart), and $(U, \varphi)$ is a coordinate neighborhood.

A smooth atlas on a topological $n$-manifold $M$ is a collection of charts $\{(U_\alpha, \varphi_\alpha)\}$ such that:

1. The $U_\alpha$ cover $M$.
2. For every pair of overlapping charts, the transition map $\varphi_\beta \circ \varphi_\alpha^{-1} : \varphi_\alpha(U_\alpha \cap U_\beta) \to \varphi_\beta(U_\alpha \cap U_\beta)$ is a smooth diffeomorphism.

Two atlases are compatible if their union is also a smooth atlas. A smooth structure on $M$ is a maximal smooth atlas.

Example 1. $\mathbb{R}^n$ with the identity chart is a smooth manifold.

Example 2. $S^n = \{x \in \mathbb{R}^{n+1} : \|x\| = 1\}$ is a smooth manifold. Use stereographic projection or the $2(n+1)$ hemisphere charts.

Example 3. The general linear group $GL_n(\mathbb{R}) = \{A \in M_n(\mathbb{R}) : \det(A) \neq 0\}$ is an open subset of $\mathbb{R}^{n^2}$, hence a smooth manifold of dimension $n^2$.

Example 4. The real projective space $\mathbb{RP}^n$ is a smooth manifold of dimension $n$.

A map $f : M \to N$ between smooth manifolds is smooth if for every $p \in M$, there exist charts $(U, \varphi)$ near $p$ and $(V, \psi)$ near $f(p)$ with $f(U) \subseteq V$, such that $\psi \circ f \circ \varphi^{-1}$ is smooth as a map between open subsets of Euclidean spaces.

A diffeomorphism is a smooth bijection with smooth inverse. If $M$ and $N$ are diffeomorphic, we write $M \cong N$.

Proposition 1.1. Diffeomorphism is an equivalence relation on the class of smooth manifolds.

For a smooth manifold $M$ of dimension $n$, the tangent space $T_p M$ at $p \in M$ can be defined in several equivalent ways:

Definition (Derivations). A tangent vector at $p$ is a linear map $v : C^\infty(M) \to \mathbb{R}$ satisfying the Leibniz rule:

$v(fg) = f(p)\, v(g) + v(f)\, g(p)$

The space of all such derivations is $T_p M$, an $n$-dimensional vector space.

Definition (Curves). A tangent vector is an equivalence class of smooth curves $\gamma : (-\varepsilon, \varepsilon) \to M$ with $\gamma(0) = p$, where $\gamma_1 \sim \gamma_2$ if they have the same derivative in any chart.

In local coordinates $(x^1, \ldots, x^n)$, a basis for $T_p M$ is given by the partial derivative operators $\{\partial/\partial x^i|_p\}$.

The differential. For a smooth map $f : M \to N$, the pushforward or differential $df_p : T_p M \to T_{f(p)} N$ is defined by:

$df_p(v)(g) = v(g \circ f)$

for $g \in C^\infty(N)$. In coordinates, $df_p$ is represented by the Jacobian matrix.

The cotangent space $T_p^* M$ is the dual vector space to $T_p M$. Elements are called covectors or differential 1-forms at $p$.

In coordinates, the basis dual to $\{\partial/\partial x^i\}$ is $\{dx^i\}$, defined by $dx^i(\partial/\partial x^j) = \delta^i_j$.

The differential of a function $f \in C^\infty(M)$ at $p$ is the covector:

$df_p(v) = v(f)$

In coordinates: $df = \frac{\partial f}{\partial x^i}\, dx^i$.

A smooth vector field $X$ on $M$ assigns a tangent vector $X_p \in T_p M$ smoothly to each $p \in M$. In coordinates:

$X = X^i(x) \frac{\partial}{\partial x^i}$

where $X^i$ are smooth functions.

Integral curves. A curve $\gamma(t)$ is an integral curve of $X$ if $\dot{\gamma}(t) = X_{\gamma(t)}$. The flow $\phi_t$ of $X$ is a one-parameter family of diffeomorphisms.

Lie bracket. The Lie bracket of two vector fields $X, Y$ is:

$[X, Y](f) = X(Y(f)) - Y(X(f))$

In coordinates: $[X, Y]^i = X^j \partial_j Y^i - Y^j \partial_j X^i$.

Problem 1. Show that $S^1$ is a smooth manifold by constructing an atlas with two charts.

Solution. Use stereographic projection from the north and south poles. For $U_1 = S^1 \setminus \{(0, 1)\}$, $\varphi_1(x, y) = x/(1 - y)$. For $U_2 = S^1 \setminus \{(0, -1)\}$, $\varphi_2(x, y) = x/(1 + y)$. The transition map $\varphi_2 \circ \varphi_1^{-1}(t) = 1/t$ is smooth on $\mathbb{R} \setminus \{0\}$. $\blacksquare$

Problem 2. Show that $T^2 = S^1 \times S^1$ is a smooth manifold.

Problem 3. Prove that the tangent bundle $TM = \bigcup_{p \in M} T_p M$ is itself a smooth $2n$-dimensional manifold.

A manifold is a space that looks like Euclidean space when you zoom in closely enough. The surface of the Earth is a two-dimensional manifold: locally it appears flat (which is why map projections work for small regions), but globally it has a different topology (it is a sphere). The key idea is that you can cover the manifold with overlapping coordinate charts, each mapping a small patch to $\mathbb{R}^n$, and the transition maps between overlapping charts are smooth.

A smooth structure on a manifold is the collection of charts that make all transition maps smooth. This is what allows you to do calculus on manifolds: derivatives, integrals, and differential equations make sense because each chart locally looks like Euclidean space. The transition maps ensure that these local constructions are compatible across overlapping charts. Partitions of unity are the technical tool that lets you glue local constructions (like Riemannian metrics or differential forms) into global ones. Manifolds are the natural setting for general relativity (spacetime is a 4-dimensional Lorentzian manifold), gauge theory (connections on principal bundles), and many areas of modern geometry and physics.

Definition. A subset $N \subseteq M$ is an embedded submanifold of dimension $k \leq n$ if for every $p \in N$, there exists a chart $(U, \varphi)$ of $M$ such that $\varphi(U \cap N) = \varphi(U) \cap (\mathbb{R}^k \times \{0\})$.

Example. $S^{n-1} \subseteq \mathbb{R}^n$ is an embedded submanifold of dimension $n-1$.

Example. The torus $T^2 \subseteq \mathbb{R}^3$ is an embedded submanifold of dimension $2$.

Theorem 1.2. Every smooth manifold admits a partition of unity: a collection of smooth functions $\{\rho_\alpha\}$ such that $\mathrm{supp}(\rho_\alpha)$ is locally finite, each $\rho_\alpha \geq 0$, and $\sum_\alpha \rho_\alpha = 1$.

Partitions of unity are used to construct global objects (Riemannian metrics, connections) from local data.

Problem 4. Show that $S^1 \times S^1$ is diffeomorphic to the torus $T^2$ embedded in $\mathbb{R}^3$.

Problem 5. Construct an atlas for $\mathbb{RP}^2$ and verify that the transition maps are smooth.

Mistake 1: Confusing a chart with an atlas A chart $(U, \varphi)$ maps a single open set to $\mathbb{R}^n$, while an atlas is a collection of charts that cover the entire manifold. A single chart is not sufficient for most manifolds (e.g., $S^1$ requires at least two charts). An atlas must have smooth transition maps on all overlaps.

Mistake 2: Assuming embedded submanifolds are the same as immersed submanifolds An embedded submanifold has the subspace topology, while an immersed submanifold may have a finer topology. For example, a dense line on a torus is an immersed submanifold but not embedded. The key difference is that embedded submanifolds are locally flat in the ambient manifold, while immersed submanifolds may self-intersect globally.

Mistake 3: Forgetting that partitions of unity are necessary for gluing local constructions Local objects like Riemannian metrics or differential forms can be defined on each chart, but gluing them into a global object requires a partition of unity. Without it, there is no canonical way to combine local data that agree on overlaps. Partitions of unity always exist on smooth manifolds, which is a key reason smoothness is assumed.

flowchart TD
    A[1_Smooth Manifolds] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

• Differential Forms: Differential forms are sections of exterior powers of the cotangent bundle on a smooth manifold.

• Riemannian Geometry: A Riemannian metric assigns an inner product to each tangent space of a smooth manifold.

• Vector Fields and Flows: Vector fields assign a tangent vector to each point of a manifold and generate flows via integral curves.

• Quantum Mechanics

• Graph Theory

• Classical Mechanics

• Electromagnetism