Smooth Manifolds

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.

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.