4 Tangent Spaces

Definition 4.1 (Smoothness in manifolds) A function $f: U \to N$ defined on an open set of a manifold $M$ is smooth at $p \in U$ if there are charts $(V, \phi), (W, \psi)$ centered at $p, f(p)$ resp. such that \[ \psi \circ f \circ \phi^{-1}: \phi(V \cap U \cap f^{-1}(W)) \to \mathbb R^n \] is smooth at $p$ (in half spaces)

A function is smooth if it is smooth everywhere defined.

Proposition 4.1 Let $M, N, R$ be three manifolds, and $U, V$ open sets of $M, N$ resp. If functions \[ f: U \to N, \quad g: V \to R \] are defined such that for some point $p \in U$,

$f(p) \in V$
$f$ is smooth at $p$ and $g$ is smooth at $f(p)$

Then $g \circ f$ is smooth at $p$.

Definition 4.2 (Germs) Suppose $M$ is a manifold, and $p \in M$. The germs of $M$ at $p$ are defined as equivalence classes specified on the set of all functionals defined on a neighborhood of $p$ such that

$f \sim g$ if they agrees on a (smaller) neighborhood of $p$.

The set of all germs of $M$ at $p$ is denoted as $C^{\infty}_p(M)$.

Definition 4.3 In similar fashion, define $C^\infty_p(M, N)$ as equivalance classes on the set of all smooth functions defined on a neighborhood of $p \in M$ whose ranges are contained in $N$ with the equivalence relation that

$f \sim g$ if they agrees on a (smaller) neighborhood of $p$.

Proposition 4.2 The germs $C^{\infty}_p(M)$ of a manifold at a point is a ring.

Definition 4.4 (Tangent vectors) A derivation or tangent vector at a point $p \in M$ in a manifold is any function $D: C_p^\infty(M) \to \mathbb R$ such that \[ D(fg) = (Df)g(p) + f(p)(Dg) \]

The set of all tangent vectors at $p$ is called its tangent space, denoted by $T_pM$.

Proposition 4.3 Tangent space at any point in any manifold is a vector space.

Definition 4.5 (Differential) Let $F \in C^\infty_p(M, N)$ be a smooth function defined on an open subset of a manifold $M$. If $F$ is smooth at $p$, then its differential at $p$ is defined as the map \[ F_*: T_pM \to T_{f(p)}N \] such that \[ F_*(X)(f) = X(f \circ F) \] for all $f \in C^\infty_{F(p)}(N)$.

Proposition 4.4 Differentials are linear transformations between tangent spaces.

Theorem 4.1 (Chain rule) If $F \in C^\infty_p(M, N), G \in C^\infty_f(p)(N, R)$, then \[ (F \circ G)_* = F_* \circ G_* \]

4.1 Bases for Tangent Spaces

Definition 4.6 (Partial derivative) If $M$ is a manifold, $p \in M$ is a point, and $\phi: U \to \mathbb R^n$ is a chart centered at $p$. Write \[ \phi = (x^1, \dots, x^n) \] where $x^i: U \to \mathbb R^n$. Define \[ \frac{\partial}{\partial x^i} \in T_pM \] with \[ \frac{\partial}{\partial x^i}f = \left . \frac{\partial(f \circ \phi^{-1})}{\partial r^i} \right \rvert_{\phi(p)} \] where the right hand side is the usual partial derivative of functions defined in euclidean spaces.

Theorem 4.2 Let $M$ be a manifold, and $\phi = (x^1, \dots, x^n)$ be a chart at $p \in M$. The partial derivatives \[ \frac{\partial}{\partial x^1}, \quad \dots, \quad \frac{\partial}{\partial x^n} \] make up a basis for $T_pM$.

Proof. It suffices to prove $T\mathbb H^n$ has dimension $n$, or that $T\mathbb H^n \simeq R^n$. Apply taylor’s expansion with remainder on any $f \in C^\infty_p(\mathbb R^n)$ with \[ f(x) = f(p) + \sum (x^i - p^i) g_i(x) \] where $g_i(p) = \left . \frac{\partial f}{\partial x^i} \right \rvert_p$.

Proposition 4.5 If $F \in C^\infty_p(M, N)$, then in local coordinates,

\[ F_*\left (\frac{\partial}{\partial x^i} \right) = \sum \frac{\partial F^i}{\partial x^j} \frac{\partial}{\partial y^i} \] where $F^i = y^i \circ F$.

4.2 Tangent Bundle

Definition 4.7 (Tangent bundle) The disjoint union of all tangent spaces of a manifold is called its tangent bundle: \[ TM = \bigsqcup T_pM = \bigcup {p} \times T_pM \]

The following defines tangent bundle as a topological space.

If $(U, \phi)$ is a chart of $M$, define \[ TU = \bigcup_{p \in U} T_pM \]
The function $\bar\phi: TU \to \phi(U) \times \mathbb R^n$ defined with \[ \bar \phi(p, v) = (\phi(p), c^1, \dots, c^n) \] where \[ v = \sum c^i \left . \frac{\partial}{\partial x^i} \right \rvert_p \]
Transfer the topology from $U$ to $TU$, making $\bar \phi$ a homeomorphism
Define the topology on $TM$ using \[ \mathcal B = \{ B: \text{$B$ is open in $TU$ for some local coordinate $U \subset M$ } \} \] as its basis.

The following defines tangent bundle as a smooth manifold.

If $\{(U, \phi)\}$ is an atlas for $M$, then $\{TU, \bar \phi\}$ is an atlas for $TM$.