The tangent bundle $T X \to X$ of a space $X$ is a bundle over $X$ whose fiber over a point $x \in X$ is a collection of infinitesimal curves in $X$ emanating at $x$: the linear approximation of $X$ at $x$.
For nice enough spaces such as differentiable manifolds or more generally microlinear spaces, the tangent bundle of $X$ is a vector bundle over $X$.
With a notion of tangent bundle comes the following terminology
A tangent vector on $X$ at $x \in X$ is an element of $T_x X$.
The tangent space of $X$ at a point $x$ is the fiber $T_x(X)$ of $T_*(X)$ over $x$;.
A tangent vector field on $X$ is a section of $T X$.
The precise definition of tangent bundle depends on the nature of the ambient category of spaces. Below we give first the traditional definitions in ordinary differential geometry. Then we discuss the construction in more general context of smooth toposes in synthetic differential geometry and other categories of generalized smooth spaces.
Here we define the notion of tangent bundle in the category Diff of smooth manifolds
There are 3 standard definitions of tangent vector known as algebraic (derivation), geometric (equivalence class of germs of curves) and physical definition (via components in local coordinate system with prescribed behaviour under change of coordinates).
Algebraically, we may define a tangent vector $v$ at $a$ on $X$ as a scalar-valued derivation on the space of germs of differentiable functions defined on $X$ near $a$, augmented by evaluation at $a$. That is, given partial functions $f$ and $g$, each defined on some neighbourhood of $a$, we have:
In light of (4), (3) is equivalent to:
Globally, we may define a tangent vector field $v$ as an ordinary (unaugmented) derivation on the space of differentiable functions defined on all of $X$. (This works for differentiable manifolds and smooth manifolds, but not for analytic manifolds and algebraic manifolds; we need to be able to change functions locally.) That is, given functions $f$ and $g$, we have:
In light of (4), (3) is again equivalent to:
Given a differentiable curve $c: \mathbf{R} \to X$, the derivative $\dot{c}$ of $c$ is a curve in the tangent bundle; given an argument $t$ and a function $f$ defined near $c(t)$, the action is given by
where $'$ indicates the usual derivative on the real line, so that $\dot{c}(t)$ is a tangent vector at $c(t)$. (We really only need $c$ to be defined on a neighbourhood of $t$, of course.)
One can also define tangent vectors at $a \in X$ to be equivalence classes of smooth curves $c : \mathbb{R} \to X$ such that $c(0) = a$, where two curves are taken to be equivalent if their first derivative coincides at $0$.
(Of course, $0$ could be replaced by any argument $t$ in this definition.)
A particularly important case is when $c$ is a level curve in some system of local coordinates $(x^1,\ldots,x^n)$ at $a$; that is, $c^i(t)$ is the point whose $i$th coordinate is $t$ and whose other coordinates are the same as at $a$. The local tangent vector field given by these curves may be written $\partial/\partial{x^i}$ or $\partial_i$ (note the placement of the scripts). This is because, if a function $f$ defined on that coordinate patch is identified with a function $f(x^1,\ldots,x^n)$ of $n$ real variables, then $\partial_i f$ becomes identified with the partial derivative $\partial{f(x^1,\ldots,x^n)}/\partial{x^i}$. In general, a local vector field $v$ on such a coordinate patch can be written
This fact can also be turned into a definition of tanget vector.
(Yet another possible definition comes from the duality with the cotangent bundle; of course, then you have to pick a definition of that that doesn't use duality.)
Note that $\partial/\partial{f}$ doesn't make sense for an arbitrary function $f$; it only makes sense when $f$ is given as one component $x^i$ of a coordinate system. That is, if $(f,g)$ and $(f,h)$ are both coordinate systems, then the two meanings of $\partial/\partial{f}$ need not be the same, because constant $g$ and constant $h$ aren't the same condition. However, we can use the more complicated notation $(\partial/\partial{f})_g$ or $(\partial/\partial{f})_h$; this is common when $X$ is a phase space in thermodynamics. Of course, if a coordinate system is fixed by convention, then there is no ambiguity.
The above definitions in ordinary differential geometry suggest the slogan
Tangent vectors are infinitesimal curves in a space.
One of the central motivations for synthetic differential geometry is the desire to provide a context in which this slogan becomes literally formally true.
(tangent bundle in smooth toposes)
Let $(\mathcal{T},(R,+,\cdot))$ be a smooth topos and write $D = \{\epsilon \in R| \epsilon^2 = 0\}$ for the standard infinitesimal interval. For $X \in \mathcal{T}$ any object (any space in $\mathcal{T}$), the tangent bundle of $X$ is the morphism
with
$T X \coloneq X^D$ the internal hom of $D$ into $X$;
$p = ev_0$ the evaluation map at the origin of $D$
$ev_0 : (U \stackrel{v}{\to} X^D) \mapsto (U \times {*} \stackrel{Id \times 0}{\to} U \times D \stackrel{\bar v}{\to} X)$,
where $\bar v$ is the hom-adjunct of $v$.
This definition captures elegantly and usefully the notion of tangent vectors as infinitesimal curves. But it is not guaranteed that the fibers of a synthetic tangent bundle $X^D$ are fiberwise linear, i.e. are fiberwise $R$-modules the way one expects. Objects $X$ for which this is true are microlinear spaces in $\mathcal{T}$. See there for more details.
A smooth topos $\mathcal{T}$ is called a well-adapted model for synthetic differential geometry if there is a full and faithful embedding Diff $\hookrightarrow \mathcal{T}$ of the cageory of manifolds into $\mathcal{T}$.
Typically, for well adapted models, under this embedding
manifolds are microlinear spaces
the synthetic definition of tangent bundle $X^D$ for $X$ a manifold does coincide with the ordinary notion of $T X$.
Let $\mathbb{L} = (C^\infty Ring^{fin})^{op}$ be the category of smooth loci. For $M$ a manifold, the exponential $M^D$ does exist in $\mathbb{L}$ and is isomorphic to the ordinary tangent bundle $T X$ of $X$. (For instance MSIA, chapter II, prop 1.12.
There are well-adapted smooth toposes $\mathcal{Z}$ and $\mathcal{B}$ presented as categories of sheaves on $\mathbb{L}$: the first for the Grothendieck topology where covers are finite open covers, the second where covers are finite open covers and projections (MSIA, chapter VI). Both topologies are subcanonical, hence the Yoneda embedding $Y : \mathbb{L} \to Sh(\mathbb{L})$ does preserve the above property.
Hence in these models for $X \in Diff$ a manifold, $T X \in Diff$ its ordinary tangent bundle and $s : Diff \to Sh(\mathbb{L})$ the full and faithful embedding, we have isomorphisms
which respect the bundle maps.
The tangent bundle of a manifold $X$ may be interpreted as a supermanifold in which $X$ has degree $0$ and the tangent vectors have degree $1$. See shifted tangent bundle.
There are useful categories of generalized smooth spaces which are neither categories of manifolds nor smooth toposes modeling synthetic differential geometry. But most of them admit useful notions of tangent bundles, too, sometimes more than one.
See Frölicher space and diffeological space for the definitions in their context.
Examples of sequences of local structures
geometry | point | first order infinitesimal | $\subset$ | formal = arbitrary order infinitesimal | $\subset$ | local = stalkwise | $\subset$ | finite |
---|---|---|---|---|---|---|---|---|
$\leftarrow$ differentiation | integration $\to$ | |||||||
smooth functions | derivative | Taylor series | germ | smooth function | ||||
curve (path) | tangent vector | jet | germ of curve | curve | ||||
smooth space | infinitesimal neighbourhood | formal neighbourhood | open neighbourhood | |||||
function algebra | square-0 ring extension | nilpotent ring extension/formal completion | ring extension | |||||
arithmetic geometry | $\mathbb{F}_p$ finite field | $\mathbb{Z}_p$ p-adic integers | $\mathbb{Z}_{(p)}$ localization at (p) | $\mathbb{Z}$ integers | ||||
Lie theory | Lie algebra | formal group | local Lie group | Lie group | ||||
symplectic geometry | Poisson manifold | formal deformation quantization | local strict deformation quantization | strict deformation quantization |
A textbook account of tangent bundles in the context of synthetic differential geometry is in
Further discussion of axiomatizations in this context is in
Discussion for diffeological spaces is in