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 with or without boundary. 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. 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, a total of \(2n\) 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, together with its
boundary version
when \(M\) has boundary.
Proposition: The Tangent Bundle Is a Smooth Manifold
For any smooth \(n\)-manifold \(M\), with or without boundary, the tangent bundle \(TM\) has a natural
topology and smooth structure making it a \(2n\)-dimensional smooth manifold with or without boundary, with
\(\partial(TM) = \pi^{-1}(\partial M)\). With respect to this structure 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. By 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.
Every point of \(M\) lies in a smooth coordinate domain, so those domains form an open cover of
\(M\). Since \(M\) is second-countable, this cover
admits a countable subcover
\(\{U_i\}\), and the preimages
\(\{\pi^{-1}(U_i)\}\) form a countable cover of \(TM\) by sets carrying these charts, so the conditions of the chart lemma are met.
For the Hausdorff condition, two tangent vectors in the same fiber lie in a common chart. For vectors
\((p, v)\) and \((q, w)\) in different fibers, the Hausdorff property of \(M\) supplies disjoint
neighborhoods of \(p\) and \(q\), and shrinking these to coordinate domains \(U \ni p\) and \(V \ni q\) makes
the preimages \(\pi^{-1}(U)\) and \(\pi^{-1}(V)\) 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 case with boundary.
Suppose now that \(M\) has boundary, and let \((U, \varphi)\) be a smooth boundary chart, so that
\(\varphi(U)\) is open in \(\mathbb{H}^n\). The set \(\varphi(U) \times \mathbb{R}^n\) is not open in
\(\mathbb{H}^{2n}\), because the half-space coordinate \(x^n\) sits in the middle of the list. Listing the
fiber components first repairs this. Define the natural chart of a boundary chart by
\[
\widetilde\varphi\!\left( v^i\,\frac{\partial}{\partial x^i}\bigg|_p \right)
= \bigl(v^1, \dots, v^n,\, x^1(p), \dots, x^n(p)\bigr),
\]
whose image \(\mathbb{R}^n \times \varphi(U)\) is an open subset of \(\mathbb{H}^{2n}\), the half-space
coordinate now being the last one. This is the identification
\(\mathbb{R}^n \times \mathbb{H}^n \cong \mathbb{H}^{2n}\) already used for
products with a single boundary factor.
Interior charts of \(M\) keep the natural chart defined above, with image open in \(\mathbb{R}^{2n}\). The
coordinate vectors at a boundary point, and hence the components \(v^i\), are the ones furnished by the
coordinate basis,
which is defined at boundary points through the identification of \(T_{\widehat p}\mathbb{H}^n\) with
\(T_{\widehat p}\mathbb{R}^n\).
We verify the hypotheses of the boundary chart lemma. Bijectivity onto an open subset of the model space is
as before, with \(\mathbb{H}^{2n}\) as the model space of a boundary chart. The image of an overlap is
\(\mathbb{R}^n \times \varphi(U \cap V)\), open in \(\mathbb{H}^{2n}\) because \(\varphi(U \cap V)\) is open
in \(\varphi(U)\), hence in \(\mathbb{H}^n\). The transition map is the same formula as in the boundaryless
case, up to the reordering of the two blocks of coordinates, and it is smooth in the half-space sense. The
transition \(\psi \circ \varphi^{-1}\) of \(M\) is smooth in the
half-space sense,
so near each point it agrees with a smooth map \(\tau\) on a Euclidean neighborhood, and the map
\((v, x) \mapsto \bigl(D\tau(x)\, v,\ \tau(x)\bigr)\) is then a smooth extension of the transition on
\(TM\) across the hyperplane. Its restriction is the transition because the Jacobian of the transition at a
boundary point, the matrix that the component transformation law uses, is the Jacobian of any smooth extension.
The countability and Hausdorff conditions are verified word for word as before, since they use only the
topology of \(M\). The boundary chart lemma therefore makes \(TM\) a smooth \(2n\)-manifold with boundary,
and the projection is smooth because its representation \((v, x) \mapsto x\) is the restriction of a linear
map.
For the boundary, the
topological invariance of the boundary
lets a single chart decide. A point \(v \in TM\) lies in \(\partial(TM)\) exactly when a (equivalently, every)
natural chart built from a smooth chart of \(M\) at \(\pi(v)\) is a boundary chart carrying \(v\) into
\(\partial\mathbb{H}^{2n}\). For the natural chart of an interior chart this never happens, and for that of a
boundary chart it means \(x^n(\pi(v)) = 0\). The same theorem applied to \(M\) says that \(\pi(v) \in \partial M\)
exactly when that chart of \(M\) is a boundary chart with \(x^n(\pi(v)) = 0\), which is the condition just
found. This gives the identity \(\partial(TM) = \pi^{-1}(\partial M)\). When
\(\partial M = \varnothing\) no boundary chart occurs and the construction reduces to the one given first.
The coordinates \((x^i, v^i)\) furnished by these charts are called the natural coordinates on
\(TM\) associated with the chart \((U, \varphi)\) of \(M\). On a boundary chart the natural chart lists the fiber
components first. The reordering is a linear isomorphism of \(\mathbb{R}^{2n}\) and changes no computation, so
we write natural coordinates as \((x^i, v^i)\) in every case and remember the ordering only when a chart must be
regarded as a map into \(\mathbb{H}^{2n}\).
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 \(\varphi(M) \times \mathbb{R}^n\), up to
the reordering of blocks in the boundary case, 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 \(TM \cong M \times \mathbb{R}^n\), where the product carries the smooth structure of a
product with a single boundary factor
when \(M\) has boundary and the ordinary product structure otherwise.
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. The pointwise
linearization becomes a smooth map.
Definition: The Global Differential
Let \(F : M \to N\) be a smooth map between smooth manifolds with or without boundary. 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 between smooth manifolds with or without boundary. 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. By the
coordinate formula for the differential,
that action is multiplication by the Jacobian of \(\widehat F\). Both parts are smooth functions of \((x, v)\),
so \(dF\) is smooth. When \(M\) or \(N\) has boundary the same representation
\((x, v) \mapsto \bigl(\widehat F(x), D\widehat F(x)\, v\bigr)\) is smooth, in the half-space sense where its
domain is not open in Euclidean space, since \(\widehat F\) extends smoothly across the hyperplane and the
extension's derivative extends the Jacobian, exactly as for the transition maps in the construction of \(TM\).
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. For (c), the
pointwise inverses assemble into \(d(F^{-1})\), which is smooth by the first part of the proof applied to
\(F^{-1}\), so \(dF\) is a diffeomorphism.
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 page 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, settles 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
\(J \subseteq \mathbb{R}\), which may be open, closed, or half-open. An interval with one or two endpoints is a
smooth \(1\)-manifold with boundary, its boundary charts being restrictions of \(t \mapsto t - a\) and
\(t \mapsto b - t\) onto open subsets of \(\mathbb{H}^1 = [0, \infty)\). Smoothness of \(\gamma\) at an
endpoint is read in the
half-space sense,
so the coordinate representation of \(\gamma\) near the endpoint extends to a smooth map on an open interval
around it. Such a curve is parametrized, and it
carries not just an image but a rate of traversal. The standard coordinate vector \(d/dt|_{t_0}\) spans the
one-dimensional tangent space \(T_{t_0}J\), which at an endpoint is identified with \(T_{t_0}\mathbb{R}\) through
the
tangent space of a half-space at a boundary point,
and pushing it forward by \(\gamma\) produces the velocity. We write \(d/dt\) rather than \(\partial/\partial t\)
by the usual convention for a single variable.
Definition: Velocity of a Curve
Let \(M\) be a smooth manifold with or without boundary, let \(\gamma : J \to M\) be a smooth curve, and let
\(t_0 \in J\), an endpoint of \(J\) or not. 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. When \(t_0\) is an endpoint of \(J\), the derivatives \(d\gamma^i/dt(t_0)\) are
one-sided. They are the derivatives at \(t_0\) of any smooth extension of the component functions,
which exists by the half-space reading of smoothness. Each is a limit of difference quotients taken inside
\(J\), so its value does not depend on the extension chosen.
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
\(\varepsilon \gt 0\), 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 \(v^n \gt 0\), we
take \([0, \varepsilon)\). If \(v^n \lt 0\), 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.
One point of bookkeeping remains. Applied to \(G\) at \(I\), the proposition produces a curve
\(\gamma\) defined only on a small
interval \((-\varepsilon, \varepsilon)\), whereas the definition of \(\mathfrak{g}\) quantifies over
curves defined on all of \(\mathbb{R}\). Precomposing with \(\sigma(t) = \varepsilon \tanh(t / \varepsilon)\),
a smooth map of \(\mathbb{R}\) into \((-\varepsilon, \varepsilon)\) with \(\sigma(0) = 0\) and
\(\sigma'(0) = 1\), produces a curve \(\gamma \circ \sigma : \mathbb{R} \to G\) through the identity. In a
chart the component functions of \(\gamma \circ \sigma\) are \(\gamma^i \circ \sigma\), so the ordinary
chain rule and the coordinate formula for the
velocity
give \((\gamma \circ \sigma)'(0) = \sigma'(0)\, \gamma'(0) = \gamma'(0)\).
The original definition of the Lie algebra is therefore exactly correct as stated, now resting on solid
ground. Here \(\mathfrak{g}\) 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 between smooth manifolds with or without boundary 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 between smooth manifolds with or without boundary, \(p \in M\), 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 because 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 identified the Lie algebras of 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. The vector \(X\) is the
velocity at the identity of the smooth curve \(\gamma(t) = e^{tX}\), an element of \(T_I G = \mathfrak{g}\).
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.
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 is in canonical one-to-one correspondence with \(T_pM\), and each has a setting in which
it is the natural choice. The pages ahead draw on all three. We describe them in turn, indicating how each is shown
equivalent to the derivation definition and where each reappears later. The equivalences are stated as sketches
with references to the results that close them, so that no step is left to an outside reference. Throughout this
section, \(M\) is a smooth manifold with or without boundary and \(p \in M\).
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 \(C^\infty_p(M)\).
The resulting correspondence \(T_pM \to \mathcal{D}_pM\) is linear. It is injective because a global derivation
that descends to the zero germ derivation annihilates every global function and is therefore itself zero.
Surjectivity holds because a germ derivation can be evaluated on global functions through their germs. Every
germ is in turn the germ of a global smooth function, since the
extension lemma
extends a representative from a closed neighborhood of \(p\) inside its domain to all of \(M\). No germ
derivation is therefore missed, and the two spaces are 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. On a connected
real-analytic or complex-analytic manifold, an analytic function that vanishes on a nonempty open set
vanishes identically, so the bump functions used for locality have no analytic counterparts, and on
real-analytic and complex-analytic manifolds the germ definition is the standard one. 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 on this page, 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.