The Tangent Bundle

The Tangent Bundle Velocity Vectors of Curves Alternative Definitions

The Tangent Bundle

Having built a tangent space at each point, we now assemble them. Gathering the tangent spaces of a manifold into a single set is straightforward; the substantive discovery is that this set is not merely an indexed family of vector spaces but a smooth manifold in its own right, of twice the dimension, on which the original projection back to \(M\) is a smooth map. This object — the tangent bundle — is where vector fields live, where flows are integrated, and where the global geometry of \(M\) first becomes visible; it is also the prototype of the vector bundles that organize all of differential geometry.

The tangent bundle as a set

Definition: The Tangent Bundle

Let \(M\) be a smooth manifold. The tangent bundle of \(M\) is the disjoint union of all its tangent spaces, \[ TM = \coprod_{p \in M} T_pM, \] whose elements we write as pairs \((p, v)\) with \(p \in M\) and \(v \in T_pM\). The projection \(\pi : TM \to M\) sends a tangent vector to the point at which it is based, \(\pi(p, v) = p\). When the base point is clear we abbreviate \((p, v)\) to \(v\) and write \(v_p\) when we wish to emphasize it.

The disjoint union is essential: a tangent vector carries its base point as part of its identity, so that vectors at different points are never identified even when they have the same components in some chart. As a first example, the Euclidean tangent bundle is a product. Since \(T_a\mathbb{R}^n \cong \mathbb{R}^n\) canonically at every point, collecting these identifications gives \(T\mathbb{R}^n \cong \mathbb{R}^n \times \mathbb{R}^n\), with the projection \(\pi\) becoming the projection onto the first factor. We will see that this product structure is special to flat spaces and a few other manifolds; in general \(TM\) is not a product, and detecting when it fails to be one is a recurring theme of the subject.

The smooth structure on \(TM\)

The charts of \(M\) induce charts on \(TM\) in the obvious way: over a coordinate domain, a tangent vector is determined by the coordinates of its base point together with its components, giving \(2n\) numbers. Verifying that these fit together into a smooth structure is an application of the smooth manifold chart lemma, the same tool that built the Grassmannian from charts.

Proposition: The Tangent Bundle Is a Smooth Manifold

For any smooth \(n\)-manifold \(M\), the tangent bundle \(TM\) has a natural topology and smooth structure making it a \(2n\)-dimensional smooth manifold, with respect to which the projection \(\pi : TM \to M\) is smooth.

Proof:

We first construct the charts. Given a smooth chart \((U, \varphi)\) for \(M\) with coordinate functions \((x^1, \dots, x^n)\), the preimage \(\pi^{-1}(U) \subseteq TM\) consists of all tangent vectors based at points of \(U\). Define \(\widetilde\varphi : \pi^{-1}(U) \to \mathbb{R}^{2n}\) by recording the base-point coordinates and the components in the coordinate basis, \[ \widetilde\varphi\!\left( v^i\,\frac{\partial}{\partial x^i}\bigg|_p \right) = \bigl(x^1(p), \dots, x^n(p),\, v^1, \dots, v^n\bigr). \] Its image is \(\varphi(U) \times \mathbb{R}^n\), an open subset of \(\mathbb{R}^{2n}\), and it is a bijection onto its image because its inverse is given explicitly by \[ \widetilde\varphi^{-1}(x^1, \dots, x^n, v^1, \dots, v^n) = v^i\,\frac{\partial}{\partial x^i}\bigg|_{\varphi^{-1}(x)} . \]

Now let \((U, \varphi)\) and \((V, \psi)\) be two smooth charts for \(M\), with corresponding charts \((\pi^{-1}(U), \widetilde\varphi)\) and \((\pi^{-1}(V), \widetilde\psi)\) on \(TM\). The sets \(\widetilde\varphi\bigl(\pi^{-1}(U) \cap \pi^{-1}(V)\bigr) = \varphi(U \cap V) \times \mathbb{R}^n\) and likewise for \(\widetilde\psi\) are open in \(\mathbb{R}^{2n}\) — the base factor \(\varphi(U \cap V)\) is open in \(\mathbb{R}^n\) because \(U \cap V\) is open in \(M\) and \(\varphi\) is a homeomorphism onto its image — and using the component transformation law the transition map is \[ \begin{align*} \widetilde\psi \circ \widetilde\varphi^{-1}(x, v) = \Big( &\tilde x^1(x), \dots, \tilde x^n(x), \\\\ &\frac{\partial \tilde x^1}{\partial x^j}(x)\, v^j, \dots, \frac{\partial \tilde x^n}{\partial x^j}(x)\, v^j \Big), \end{align*} \] which is smooth, since the base-point part is the smooth transition map of \(M\) and the fiber part is polynomial in \(v\) with smooth coefficients.

Choosing a countable cover \(\{U_i\}\) of \(M\) by smooth coordinate domains yields a countable cover \(\{\pi^{-1}(U_i)\}\) of \(TM\) by sets carrying these charts, and the conditions of the chart lemma are met. For the Hausdorff condition, two tangent vectors in the same fiber lie in a common chart, while vectors \((p, v)\) and \((q, w)\) in different fibers can be separated by taking disjoint coordinate domains \(U \ni p\) and \(V \ni q\) in \(M\), whose preimages \(\pi^{-1}(U)\) and \(\pi^{-1}(V)\) are disjoint chart domains separating the two. The chart lemma therefore endows \(TM\) with a \(2n\)-dimensional smooth structure. Finally, in the charts \((U, \varphi)\) and \((\pi^{-1}(U), \widetilde\varphi)\) the projection has the coordinate representation \(\pi(x, v) = x\), which is smooth.

The coordinates \((x^i, v^i)\) furnished by these charts are called the natural coordinates on \(TM\) associated with the chart \((U, \varphi)\) of \(M\). If \(M\) has boundary, the same construction works with one cosmetic change: rearranging the natural coordinates of a boundary chart to place the fiber components first, as \((v^i, x^i)\), turns the natural chart into a boundary chart, so that \(TM\) is a smooth manifold with boundary. We will not need this refinement and treat \(M\) as boundaryless by default.

Proposition: Tangent Bundles of Single-Chart Manifolds Are Products

If \(M\) is a smooth \(n\)-manifold with or without boundary that can be covered by a single smooth chart, then \(TM\) is diffeomorphic to \(M \times \mathbb{R}^n\).

Proof:

If \((U, \varphi)\) is a global smooth chart, then \(U = M\) and \(\varphi\) is a diffeomorphism onto an open subset of \(\mathbb{R}^n\) or \(\mathbb{H}^n\). The construction above makes the natural chart \(\widetilde\varphi\) a bijection from \(TM = \pi^{-1}(M)\) onto \(\varphi(M) \times \mathbb{R}^n\), and the smooth structure on \(TM\) is defined precisely by declaring \(\widetilde\varphi\) to be a diffeomorphism. Composing with \(\varphi^{-1} \times \mathrm{Id}\) gives a diffeomorphism \(TM \cong M \times \mathbb{R}^n\).

Why "Bundle" and Not "Product"

The local picture is always a product: over any coordinate domain, \(TM\) looks like \(U \times \mathbb{R}^n\), and the single-chart case above makes the whole bundle a product. It is tempting to conclude that \(TM\) is globally \(M \times \mathbb{R}^n\) for every \(M\), but this is false for most manifolds. The local product structure can fail to extend to a global one: as the coordinate domains are glued together, the fibers are identified through the component transformation law, and the resulting twisting can obstruct any global splitting of \(TM\) as a product. The word bundle records exactly this gap between the local product structure and the global geometry that may obstruct it. The tangent bundle is the first and most important example of a vector bundle.

The global differential

A smooth map between manifolds differentiates pointwise to a family of differentials, one at each point. Assembling these into a single map between tangent bundles globalizes the differential, turning the pointwise linearization into a smooth map of \(2n\)-dimensional manifolds.

Definition: The Global Differential

Let \(F : M \to N\) be a smooth map. The global differential (or global tangent map) is the map \[ dF : TM \to TN \] whose restriction to each tangent space \(T_pM\) is the pointwise differential \(dF_p\); that is, \(dF(p, v) = (F(p), dF_p(v))\). We continue to write \(dF_p(v)\) for the action of the differential at a single point and \(dF(v)\) for the global map applied to a tangent vector \(v \in TM\).

Proposition: Properties of the Global Differential

Let \(F : M \to N\) and \(G : N \to P\) be smooth maps. The global differential \(dF : TM \to TN\) is smooth, and it satisfies

(a) \(d(G \circ F) = dG \circ dF\);

(b) \(d(\mathrm{Id}_M) = \mathrm{Id}_{TM}\);

(c) if \(F\) is a diffeomorphism, then \(dF : TM \to TN\) is a diffeomorphism, with inverse \((dF)^{-1} = d(F^{-1})\).

Proof:

For smoothness, work in natural coordinates. With respect to charts \((\pi^{-1}(U), \widetilde\varphi)\) on \(TM\) and \((\pi^{-1}(V), \widetilde\psi)\) on \(TN\), the coordinate representation of \(dF\) records the base-point map \(\widehat F\) together with the action of the differential on components, which by the coordinate formula for the differential is multiplication by the Jacobian of \(\widehat F\). Both parts are smooth functions of \((x, v)\), so \(dF\) is smooth.

Properties (a)–(c) follow immediately from the corresponding pointwise properties of the differential. Each identity holds on every fiber \(T_pM\) by the pointwise statement, and since the global differential is defined fiber by fiber, the fiberwise identities assemble into the stated global ones. For instance \(d(G \circ F)\) restricts on \(T_pM\) to \(d(G \circ F)_p = dG_{F(p)} \circ dF_p\), which is the restriction of \(dG \circ dF\); as this holds at every \(p\), the two global maps coincide.

With \(TM\) established as a smooth manifold and \(dF\) as a smooth map between such manifolds, the differential has graduated from a pointwise gadget to a global construction. The remaining sections of this chapter exploit this globalization in two directions: by following individual tangent vectors along curves, which recovers the velocity-based picture from which Lie algebras were first defined, and by re-examining what a tangent vector is from several equivalent points of view.

Velocity Vectors of Curves

The derivation definition of a tangent vector is intrinsic and clean, but it is far from the picture most people carry of a tangent vector as the velocity of a moving point. This section reconciles the two. A smooth curve in \(M\) has, at each instant, a velocity that is a tangent vector in the derivation sense; conversely every tangent vector arises this way. The result restores the geometric image and, in doing so, completes a debt left open when we first defined Lie algebras as velocities of curves through the identity.

The velocity of a curve

By a curve in \(M\) we mean a smooth map \(\gamma : J \to M\) from an interval \(J \subseteq \mathbb{R}\); it is a parametrized curve, carrying not just an image but a rate of traversal. The standard coordinate vector \(d/dt|_{t_0}\) spans the one-dimensional tangent space \(T_{t_0}\mathbb{R}\) — written \(d/dt\) rather than \(\partial/\partial t\) by the usual convention for a single variable — and pushing it forward by \(\gamma\) produces the velocity.

Definition: Velocity of a Curve

Let \(\gamma : J \to M\) be a smooth curve and \(t_0 \in J\). The velocity of \(\gamma\) at \(t_0\) is the tangent vector \[ \gamma'(t_0) = d\gamma\!\left( \frac{d}{dt}\bigg|_{t_0} \right) \in T_{\gamma(t_0)}M, \] also written \(\dot\gamma(t_0)\) or \(\tfrac{d\gamma}{dt}(t_0)\). It acts on a smooth function \(f\) by differentiating \(f\) along the curve, \[ \gamma'(t_0)\, f = (f \circ \gamma)'(t_0). \] In any chart, writing the component functions of \(\gamma\) as \(\gamma^i(t) = x^i \circ \gamma(t)\), the coordinate formula for the differential gives \[ \gamma'(t_0) = \frac{d\gamma^i}{dt}(t_0)\, \frac{\partial}{\partial x^i}\bigg|_{\gamma(t_0)}, \] so the components of the velocity are the ordinary derivatives of the component functions, exactly as in Euclidean space.

Every tangent vector is a velocity

The construction can be reversed: any tangent vector, however abstractly defined, is realized as the velocity of an explicit curve. This is the precise statement licensing the geometric picture.

Proposition: Every Tangent Vector Is a Velocity

Let \(M\) be a smooth manifold with or without boundary and \(p \in M\). Every \(v \in T_pM\) is the velocity \(\gamma'(0)\) of some smooth curve \(\gamma\) in \(M\) with \(\gamma(0) = p\).

Proof:

Suppose first that \(p\) is an interior point, which includes the boundaryless case. Let \((U, \varphi)\) be a smooth chart centered at \(p\), and write \(v = v^i\,\partial/\partial x^i|_p\). For sufficiently small \(\varepsilon > 0\), define \(\gamma : (-\varepsilon, \varepsilon) \to U\) by the coordinate representation \[ \gamma(t) = \varphi^{-1}(t v^1, \dots, t v^n). \] This is a smooth curve with \(\gamma(0) = \varphi^{-1}(0) = p\), and computing its velocity from the component formula gives \[ \gamma'(0) = \frac{d(t v^i)}{dt}\bigg|_{0}\, \frac{\partial}{\partial x^i}\bigg|_p = v^i\, \frac{\partial}{\partial x^i}\bigg|_p = v. \]

Now suppose \(p \in \partial M\), and let \((U, \varphi)\) be a smooth boundary chart centered at \(p\), with \(v = v^i\,\partial/\partial x^i|_p\) as before. The coordinate formula \(\gamma(t) = \varphi^{-1}(tv^1, \dots, tv^n)\) represents a point of \(M\) only when \(t v^n \geq 0\), since the chart maps into the half-space \(\mathbb{H}^n = \{x^n \geq 0\}\). We accommodate this by restricting the domain according to the sign of \(v^n\): if \(v^n = 0\) we take the domain \((-\varepsilon, \varepsilon)\) as before; if \(v^n > 0\) we take \([0, \varepsilon)\); and if \(v^n < 0\) we take \((-\varepsilon, 0]\). In each case the same coordinate formula defines a smooth curve in \(M\) with \(\gamma(0) = p\) and \(\gamma'(0) = v\).

Closing the Lie Algebra Definition

We can now complete the identification begun when tangent spaces were first defined. The Lie algebra of a matrix group \(G\) was introduced as the set of velocity vectors \(\gamma'(0)\) of smooth curves through the identity, \(\mathfrak{g} = \{\gamma'(0) : \gamma(0) = I\}\). At the time, both halves of that phrase were provisional: we had neither a tangent space at \(I\) to contain the velocities nor a definition of velocity itself. The derivation construction supplied the first, identifying \(\mathfrak{g}\) with \(T_I G\); the present proposition supplies the second, making \(\gamma'(0)\) a genuine tangent vector and showing that every element of \(T_I G\) is realized as such a velocity. The original definition of the Lie algebra is therefore exactly correct as stated, now resting on solid ground: \(\mathfrak{g}\) is the tangent space \(T_I G\), and its elements are precisely the velocities of curves through the identity.

Velocities under composition

Velocities transform predictably under smooth maps: pushing a curve forward by \(F\) and then taking its velocity is the same as taking the velocity and then applying the differential. This is the curve-level form of the chain rule.

Proposition: The Velocity of a Composite Curve

Let \(F : M \to N\) be a smooth map and \(\gamma : J \to M\) a smooth curve. For any \(t_0 \in J\), the velocity at \(t_0\) of the composite curve \(F \circ \gamma : J \to N\) is \[ (F \circ \gamma)'(t_0) = dF\bigl(\gamma'(t_0)\bigr). \]

Proof:

Returning to the definition of velocity and using the chain rule for differentials, \[ \begin{align*} (F \circ \gamma)'(t_0) &= d(F \circ \gamma)\!\left( \frac{d}{dt}\bigg|_{t_0} \right) \\\\ &= dF \circ d\gamma\!\left( \frac{d}{dt}\bigg|_{t_0} \right) = dF\bigl(\gamma'(t_0)\bigr). \end{align*} \]

Read in reverse, this proposition becomes a practical computational tool. To evaluate \(dF_p(v)\) one need not pass through coordinates at all: it suffices to choose any curve realizing \(v\) and differentiate the image curve.

Corollary: Computing the Differential via a Velocity Vector

Let \(F : M \to N\) be a smooth map, \(p \in M\), and \(v \in T_pM\). For any smooth curve \(\gamma\) with \(\gamma(0) = p\) and \(\gamma'(0) = v\), \[ dF_p(v) = (F \circ \gamma)'(0). \]

Proof:

Such a curve exists by the proposition that every tangent vector is a velocity, and the previous proposition gives \((F \circ \gamma)'(0) = dF(\gamma'(0)) = dF_p(v)\). In particular the left-hand side does not depend on which realizing curve \(\gamma\) is chosen, since the right-hand side \(dF_p(v)\) makes no reference to \(\gamma\).

Recovering the Exponential Picture

This corollary retroactively justifies the calculations that defined matrix Lie groups. The one-parameter subgroups \(t \mapsto e^{tX}\) were treated throughout as curves to be differentiated, with \(X\) recovered as \(\tfrac{d}{dt}\big|_0 e^{tX}\). That manipulation is now legitimate on the nose: \(X\) is the velocity at the identity of the smooth curve \(\gamma(t) = e^{tX}\), an element of \(T_I G = \mathfrak{g}\), and the corollary is the general principle that lets one read off a differential by differentiating along such a curve. The informal "differentiate the exponential" of the earlier development and the formal differential \(dF_p\) are one and the same operation.

The velocity perspective and the derivation perspective are now seen to be two views of a single object. The final section confirms that this multiplicity of viewpoints is the rule rather than the exception, surveying several equivalent definitions of the tangent space and indicating where each is put to use downstream.

Alternative Definitions of the Tangent Space

The derivation definition is one of several routes to the tangent space, and a reader consulting the literature will meet the others. Each characterizes \(T_pM\) up to canonical isomorphism, and each has a setting in which it is the natural choice; later chapters draw on all three. We describe them in turn, indicating how each is shown equivalent to the derivation definition and where each reappears downstream. The equivalences are stated as sketches with references to the results that close them, in keeping with the principle that every argument the site relies on is resolved within the site.

Tangent vectors as derivations of the space of germs

The most common alternative is built from germs of smooth functions, which localize a function to an arbitrarily small neighborhood of a point.

Definition: Germs and the Germ Algebra

A smooth function element on \(M\) is an ordered pair \((f, U)\) with \(U \subseteq M\) open and \(f : U \to \mathbb{R}\) smooth. Fixing \(p \in M\), declare two function elements whose domains contain \(p\) equivalent, \((f, U) \sim (g, V)\), if \(f \equiv g\) on some neighborhood of \(p\). The equivalence class of \((f, U)\) is the germ of \(f\) at \(p\), written \([f]_p\); the domain may be dropped from the notation, since restricting \(f\) to any neighborhood of \(p\) represents the same germ. The set of all germs at \(p\), denoted \(C^\infty_p(M)\), is a real vector space and an associative algebra under \[ \begin{align*} c\,[f]_p &= [cf]_p, \\\\ [f]_p + [g]_p &= [f + g]_p, \\\\ [f]_p\,[g]_p &= [fg]_p, \end{align*} \] each defined on the intersection of the relevant domains, with zero element the germ of the zero function.

Definition: Tangent Vectors as Germ Derivations

A derivation of \(C^\infty_p(M)\) is a linear map \(v : C^\infty_p(M) \to \mathbb{R}\) satisfying the Leibniz rule \[ v[fg]_p = f(p)\, v[g]_p + g(p)\, v[f]_p. \] The space of all such derivations is denoted \(\mathcal{D}_pM\). It is canonically isomorphic to the tangent space, \(\mathcal{D}_pM \cong T_pM\).

Proof Sketch:

A derivation at \(p\) in our sense acts on global smooth functions, but locality shows its value depends only on the germ, so it descends to a map on \(C^\infty_p(M)\); this gives a linear map \(T_pM \to \mathcal{D}_pM\). It is injective because if a global derivation descends to the zero germ derivation, then by locality it annihilates every global function and so is itself zero; and surjective because a germ derivation can be evaluated on global functions through their germs — the extension lemma guarantees every germ is the germ of a global smooth function, so no germ derivation is missed. The two spaces are thus identified.

The germ definition makes the local nature of the tangent space especially transparent, with no bump functions required. This is decisive in settings where bump functions are unavailable: since there are no analytic functions that vanish on an open set without vanishing identically, the germ definition is the only one available on real-analytic and complex-analytic manifolds. Its cost is a further layer of abstraction atop an already abstract construction.

Tangent vectors as equivalence classes of curves

A second approach formalizes the idea of two curves "having the same velocity" at a point and defines a tangent vector to be such a class of curves.

Definition: Tangent Vectors as Curve Classes

Consider all smooth curves \(\gamma : J \to M\) with \(0 \in J\) and \(\gamma(0) = p\). Declare two such curves equivalent, \(\gamma_1 \sim \gamma_2\), if \[ (f \circ \gamma_1)'(0) = (f \circ \gamma_2)'(0) \] for every smooth real-valued function \(f\) defined near \(p\). The set of equivalence classes is denoted \(V_pM\), and there is a canonical bijection \(V_pM \leftrightarrow T_pM\).

Proof Sketch:

The velocity map \([\gamma] \mapsto \gamma'(0)\) sends a curve class to a tangent vector, and it is well defined precisely because two curves are declared equivalent exactly when their velocities act identically on all functions — which is to say when \(\gamma_1'(0) = \gamma_2'(0)\). It is surjective because every tangent vector is the velocity of some curve, as shown earlier in this chapter, and injective by the very definition of the equivalence relation. The differential of a smooth map \(F : M \to N\) is then simply \([\gamma] \mapsto [F \circ \gamma]\), and the velocity at \(t_0\) of a curve \(\gamma\) is the class of the reparametrized curve \(t \mapsto \gamma(t_0 + t)\).

This definition is the most geometrically intuitive of the three. Its serious drawback is that the vector-space structure on \(V_pM\) is not at all obvious: adding two curve classes has no evident meaning until one transports the operation across the bijection from \(T_pM\). The curve picture nonetheless becomes essential when tangent spaces to submanifolds are characterized as classes of curves constrained to lie in the submanifold, and again when the velocity of a one-parameter group of motions is read directly as a curve class.

Tangent vectors as equivalence classes of \(n\)-tuples

The final approach is the most computational and, historically, the first.

Definition: Tangent Vectors as \(n\)-Tuple Classes

A tangent vector at \(p\) is a rule assigning to each smooth coordinate chart containing \(p\) an ordered \(n\)-tuple \((v^1, \dots, v^n) \in \mathbb{R}^n\), subject to the requirement that the tuples assigned to overlapping charts transform according to the component transformation law \[ \tilde v^j = \frac{\partial \tilde x^j}{\partial x^i}(\widehat p)\, v^i. \]

This is the oldest definition of all, and many physicists still think of tangent vectors this way. The correspondence with the derivation definition assigns to a tangent vector \(v\) the tuple of its components \(v^i = v(x^i)\) in each chart; the transformation law is then exactly the change-of-components rule already established, so the assignment is consistent across charts and reversible. In this approach the velocity of a curve is given by the usual Euclidean formula in coordinates, and the differential of a smooth map is determined by its Jacobian; one then has to verify, through tedious but routine computations with the chain rule, that these operations are well defined independently of the chosen coordinates. The pattern of objects-with-a-transformation-law is the prototype of the tensor, where the same idea is applied to multi-indexed arrays.

Which of these characterizations one takes as the definition is a matter of taste. The derivation definition adopted here, abstract as it first appears, has several advantages: tangent vectors are concrete objects with no equivalence classes involved, the vector-space structure on \(T_pM\) is immediate, and the differential and the velocity of a curve acquire coordinate-free definitions directly. These are the qualities that recommend it as the foundation, even as the alternative pictures remain the right language in their own settings.