Tangent Spaces and Tangent Bundles

There are several equivalent definitions of the tangent space $T_p M$ at $p \in M$:

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

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

Definition (Equivalence Classes of 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 $(\varphi \circ \gamma_1)'(0) = (\varphi \circ \gamma_2)'(0)$ in some (hence every) chart.

Proposition 2.1. $T_p M$ is a vector space of dimension $n = \dim M$.

The tangent bundle of $M$ is

$TM = \bigsqcup_{p \in M} T_p M = \{(p, v) : p \in M,\ v \in T_p M\}$

It is a smooth manifold of dimension $2n$ with projection $\pi : TM \to M$ given by $\pi(p, v) = p$.

A vector field on $M$ is a smooth section of $TM$: a smooth map $X : M \to TM$ with $\pi \circ X = \mathrm{id}_M$, written $X(p) = X_p \in T_p M$.

Let $f : M \to N$ be a smooth map. The differential (or pushforward) of $f$ at $p$ is the linear map:

$df_p : T_p M \to T_{f(p)} N, \quad df_p(v)(g) = v(g \circ f)$

for $v \in T_p M$ and $g \in C^\infty(N)$. In local coordinates, $df_p$ is represented by the Jacobian matrix $[D(f \circ \varphi^{-1})](\varphi(p))$.

Proposition 2.2 (Chain Rule). $d(g \circ f)_p = dg_{f(p)} \circ df_p$.

In local coordinates $(x^1, \ldots, x^n)$ around $p$, the coordinate vectors $\partial/\partial x^i|_p$ form a basis of $T_p M$. Any tangent vector $v \in T_p M$ can be expressed as:

$v = v^i \frac{\partial}{\partial x^i}\bigg|_p$

For a smooth function $f : M \to \mathbb{R}$, the action of $v$ on $f$ is:

$v(f) = v^i \frac{\partial f}{\partial x^i}(p)$

Example 2.1. On $\mathbb{R}^n$, $T_p \mathbb{R}^n \cong \mathbb{R}^n$ with basis $\partial/\partial x^1, \ldots, \partial/\partial x^n$. A tangent vector $v = (v^1, \ldots, v^n)$ acts on $f \in C^\infty(\mathbb{R}^n)$ by $v(f) = \sum v^i \partial f/\partial x^i(p)$.

Example 2.2. On the sphere $S^2$ at a point $p = (\theta_0, \phi_0)$, the tangent space $T_p S^2$ is spanned by $\partial/\partial\theta$ and $\partial/\partial\phi$ evaluated at $p$.

If $(x^1, \ldots, x^n)$ and $(y^1, \ldots, y^n)$ are two coordinate systems around $p$, the transition formula for tangent vectors is:

$\frac{\partial}{\partial y^j} = \frac{\partial x^i}{\partial y^j} \frac{\partial}{\partial x^i}$

Thus the components of a vector $v = v^i \partial/\partial x^i = \tilde v^j \partial/\partial y^j$ transform as:

$\tilde v^j = v^i \frac{\partial y^j}{\partial x^i}$

This contravariant transformation law characterizes tangent vectors: their components transform using the Jacobian of the coordinate change.

Example 2.3. In polar coordinates $(r, \theta)$ on $\mathbb{R}^2$, the relation between Cartesian and polar basis vectors is:

$\frac{\partial}{\partial r} = \cos\theta\,\frac{\partial}{\partial x} + \sin\theta\,\frac{\partial}{\partial y}, \quad \frac{\partial}{\partial\theta} = -r\sin\theta\,\frac{\partial}{\partial x} + r\cos\theta\,\frac{\partial}{\partial y}$

The dual space $T_p^* M = (T_p M)^*$ is called the cotangent space at $p$. Its elements are covectors (linear functionals on $T_p M$). The basis dual to $\{\partial/\partial x^i\}$ is denoted $\{dx^i|_p\}$, where:

$dx^i\left(\frac{\partial}{\partial x^j}\right) = \delta^i_j$

Proposition 2.3. For a smooth function $f : M \to \mathbb{R}$, the differential $df_p \in T_p^* M$ is given in coordinates by:

$df_p = \frac{\partial f}{\partial x^i}(p)\, dx^i|_p$

The tangent bundle $TM$ is a special case of a vector bundle: a smooth manifold $E$ with a surjective submersion $\pi : E \to M$ such that each fiber $\pi^{-1}(p)$ is a vector space, and local trivializations exist.

Definition. A vector bundle of rank $k$ over $M$ is a smooth manifold $E$ with a smooth map $\pi : E \to M$ such that for every $p \in M$ there exists a neighborhood $U$ and a diffeomorphism $\Phi : \pi^{-1}(U) \to U \times \mathbb{R}^k$ with $\pi = \mathrm{pr}_1 \circ \Phi$ and each fiber $\pi^{-1}(p)$ maps linearly to $\{p\} \times \mathbb{R}^k$.

Example 2.4. The tangent bundle $TM$ is a rank $n$ vector bundle over $M$. The cotangent bundle $T^*M$ is also a rank $n$ vector bundle, dual to $TM$.

Example 2.5. The trivial bundle $M \times \mathbb{R}^k$ is a rank $k$ vector bundle. A manifold $M$ is parallelizable if $TM \cong M \times \mathbb{R}^n$. For example, $S^1$ is parallelizable but $S^2$ is not (by the hairy ball theorem).

Problem 1. Let $M = \mathbb{R}^2$ with coordinates $(x, y)$ and let $f : \mathbb{R}^2 \to \mathbb{R}$ be $f(x, y) = x^2 + y^2$. Compute $df_{(1,0)}$ in coordinates.

Solution. $\partial f/\partial x = 2x$, $\partial f/\partial y = 2y$. At $(1,0)$, $df_{(1,0)} = 2x\, dx + 2y\, dy|_{(1,0)} = 2\, dx$. So $df_{(1,0)}(v) = 2v^1$. $\blacksquare$

Problem 2. Show that $T_p S^2$ is isomorphic to $\{v \in \mathbb{R}^3 : p \cdot v = 0\}$.

Solution. Consider the embedding $S^2 \subseteq \mathbb{R}^3$. A curve $\gamma(t)$ on $S^2$ satisfies $\gamma(t) \cdot \gamma(t) = 1$. Differentiating: $\gamma'(0) \cdot p = 0$, so every tangent vector is orthogonal to $p$. Conversely, any $v \perp p$ is tangent to the great circle in the $p$-$v$-plane. Thus $T_p S^2 \cong p^\perp$. $\blacksquare$

• Curvature: Uses the tangent bundle and connections to define the Riemann curvature tensor, measuring how parallel transport rotates vectors.

• The Gauss-Bonnet Theorem: Relates the integral of Gaussian curvature over a surface to its Euler characteristic, a topological invariant of the tangent bundle.

• Applications: Uses tangent spaces and vector fields to model geodesics as a rule relativity and configuration spaces in robotics.

• Quantum Mechanics

• Graph Theory

• Classical Mechanics

• Electromagnetism

Mistake 1: Confusing the two definitions of tangent vectors The derivation definition and the equivalence-of-curves definition are equivalent, but students often try to use the wrong one in context. When computing with coordinates, the derivation approach is cleaner. When proving geometric results about curves on manifolds, the curve-equivalence approach is more natural.

Mistake 2: Misapplying the chain rule for pushforwards The chain rule states $d(g \circ f)_p = dg_{f(p)} \circ df_p$, where the differentials compose in the order $g$ then $f$, not $f$ then $g$. Students frequently write $df_p \circ dg_{f(p)}$ or forget that the middle spaces must match: $df_p : T_pM \to T_{f(p)}N$ and $dg_{f(p)} : T_{f(p)}N \to T_{g(f(p))}P$.

Mistake 3: Forgetting that tangent bundle components transform contravariantly Under a coordinate change $y = y(x)$, the components of a tangent vector transform as $\tilde{v}^j = v^i \partial y^j/\partial x^i$, using the Jacobian matrix. Students often mistakenly apply the inverse Jacobian, which would give covariant (covector) transformation, not contravariant.

flowchart TD
    A[2_Tangent Spaces And Tangent Bundles] --> B[Key Concepts]
    A --> C[Core Principles]
    A --> D[Practical Applications]
    B --> E[Fundamental definitions]
    C --> F[Design patterns]
    D --> G[Real-world usage]

The tangent space at a point on a manifold is the collection of all possible velocities of curves passing through that point — it is the “flat approximation” to the curved space at that point. On a sphere, the tangent space at the north pole is the horizontal plane touching the sphere there. The tangent bundle glues together all tangent spaces into a single space, giving a manifold a way to talk about directions and velocities globally. A differential form is a machine that takes a tangent vector and returns a number, consistently across all points. The exterior derivative of a form measures how the form changes — it is the infinitesimal version of Stokes’ theorem.

1. Prove that the curve and derivation definitions of $T_p M$ are equivalent.
2. Compute the transition matrix for $T_p \mathbb{R}^2$ between Cartesian and polar coordinates.
3. Show that $d(fg)_p = f(p) dg_p + g(p) df_p$.
4. Prove that $T(S^1)$ is diffeomorphic to $S^1 \times \mathbb{R}$.
5. Show that if $M$ is an $n$-dimensional manifold, then $TM$ is a $2n$-dimensional manifold.

This section provides detailed coverage of advanced concepts, including full derivations, proofs, and extended examples.

Complete mathematical derivations and proofs are provided where appropriate. Each step is explained to ensure understanding of the underlying reasoning.

Advanced examples demonstrate the application of concepts to complex problems. These examples go beyond standard exam questions to develop deeper understanding.

This material connects to current research and advanced applications in the field. Understanding these connections provides context for the study material.

Ensure you have mastered the prerequisite material before attempting this advanced content.