tangent bundle


Differential geometry

differential geometry

synthetic differential geometry








The tangent bundle TXXT X \to X of a (sufficiently differentiable) space XX is a bundle over XX whose fiber over a point xXx \in X is the tangent space at that point, namely the collection of infinitesimal curves in XX emanating at xx: “tangent vectors”.

For nice enough spaces such as differentiable manifolds or more generally microlinear spaces, the tangent bundle of XX is a vector bundle over XX.

For example the graphics on the right shows the 2-sphere with one of its tangent spaces. The tangent bundle of the sphere is the union of all these tangent spaces, regarded as a topological bundle of vector space (a vector bundle) over the 2-sphere.

graphics grabbed from Hatcher

With a notion of tangent bundle comes the following terminology

  • A tangent vector on XX at xXx \in X is an element of T xXT_x X.

  • The tangent space of XX at a point xx is the fiber T x(X)T_x(X) of T *(X)T_*(X) over xx;.

  • A tangent vector field on XX is a section of TXT 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.

Definitions in ordinary differential geometry

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).

Algebraic definition

Algebraically, we may define a tangent vector vv at aa on XX as a scalar-valued derivation on the space of germs of differentiable functions defined on XX near aa, augmented by evaluation at aa. That is, given partial functions ff and gg, each defined on some neighbourhood of aa, we have:

  1. v[f]=v[g]v[f] = v[g] if f=gf = g on some (perhaps smaller) neighbourhood of aa,
  2. v[f+g]=v[f]+v[g]v[f + g] = v[f] + v[g],
  3. v[cf]=cv[f]v[c f] = c\, v[f] for cc a scalar,
  4. v[fg]=f(a)v[g]+v[f]g(a)v[f g] = f(a)\, v[g] + v[f]\, g(a);

In light of (4), (3) is equivalent to:

  • v[k]=0v[k] = 0 for kk a constant function (or indeed, for any function constant on any neighbourhood of aa).

Globally, we may define a tangent vector field vv as an ordinary (unaugmented) derivation on the space of differentiable functions defined on all of XX. (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 ff and gg, we have:

  1. v[f]=v[g]v[f] = v[g] if f=gf = g (so really, the only reason to list this is to keep the numbering the same),
  2. v[f+g]=v[f]+v[g]v[f + g] = v[f] + v[g],
  3. v[cf]=cv[f]v[c f] = c\, v[f] for cc a scalar,
  4. v[fg]=fv[g]+v[f]gv[f g] = f\, v[g] + v[f]\, g;

In light of (4), (3) is again equivalent to:

Given a differentiable curve c:RXc: \mathbf{R} \to X, the derivative c˙\dot{c} of cc is a curve in the tangent bundle; given an argument tt and a function ff defined near c(t)c(t), the action is given by

c˙[f](t)=(fc)(t), \dot{c}[f](t) = (f \circ c)'(t) ,

where ' indicates the usual derivative on the real line, so that c˙(t)\dot{c}(t) is a tangent vector at c(t)c(t). (We really only need cc to be defined on a neighbourhood of tt, of course.)

Geometric definition

One can also define tangent vectors at aXa \in X to be equivalence classes of smooth curves c:Xc : \mathbb{R} \to X such that c(0)=ac(0) = a, where two curves are taken to be equivalent if their first derivative coincides at 00.

(Of course, 00 could be replaced by any argument tt in this definition.)

A particularly important case is when cc is a level curve in some system of local coordinates (x 1,,x n)(x^1,\ldots,x^n) at aa; that is, c i(t)c^i(t) is the point whose iith coordinate is tt and whose other coordinates are the same as at aa. The local tangent vector field given by these curves may be written /x i\partial/\partial{x^i} or i\partial_i (note the placement of the scripts). This is because, if a function ff defined on that coordinate patch is identified with a function f(x 1,,x n)f(x^1,\ldots,x^n) of nn real variables, then if\partial_i f becomes identified with the partial derivative f(x 1,,x n)/x i\partial{f(x^1,\ldots,x^n)}/\partial{x^i}. In general, a local vector field vv on such a coordinate patch can be written

v= iv i i. v = \sum_i v^i\, \partial_i .

This fact can also be turned into a definition of tangent 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 /f\partial/\partial{f} doesn't make sense for an arbitrary function ff; it only makes sense when ff is given as one component x ix^i of a coordinate system. That is, if (f,g)(f,g) and (f,h)(f,h) are both coordinate systems, then the two meanings of /f\partial/\partial{f} need not be the same, because constant gg and constant hh aren't the same condition. However, we can use the more complicated notation (/f) g(\partial/\partial{f})_g or (/f) h(\partial/\partial{f})_h; this is common when XX is a phase space in thermodynamics. Of course, if a coordinate system is fixed by convention, then there is no ambiguity.

Definition by gluing construction

If MM is a smooth nn-manifold, then we know the tangent space at each point is isomorphic to n\mathbb{R}^n. We can exploit this fact to construct the tangent space by gluing many copies of n\mathbb{R}^n together. However, one drawback is that it is not immediately obvious that the result is independent of the atlas chosen.

Let MM be a smooth nn-manifold defined by an atlas (U α,ϕ α)(U_\alpha, \phi_\alpha). Then we may define its tangent bundle TMT M by a gluing construction in TopTop, taking TMT M to be the quotient of the disjoint sum

αU α× n\sum_\alpha U_\alpha \times \mathbb{R}^n

obtained by dividing by the equivalence relation

(pU α,v)(pU β,g αβ(p)v)(p \in U_\alpha, v) \sim (p \in U_\beta, g_{\alpha\beta}(p) v)

where pU αU βp \in U_\alpha \cap U_\beta, and g αβ(p)GL( n)g_{\alpha\beta}(p) \in GL(\mathbb{R}^n) is the result of differentiating the transition function ϕ αβ\phi_{\alpha\beta} at the point ϕ α(p)\phi_\alpha(p). We thus obtain a covering U α× nU_\alpha \times \mathbb{R}^n of TMT M, and these form coordinate charts of a smooth manifold structure on TMT M in a more or less evident way, and the projection map is given by

p:(x,v)x. p: (x, v) \mapsto x.

In the above construction, the charts ϕ α\phi_\alpha were not used directly; Instead, the transition maps

g αβ:U αU βGL( n)g_{\alpha \beta}: U_\alpha \cap U_\beta \to GL(\mathbb{R}^n)

are what were needed to construct the tangent bundle. These maps satisfy Čech 1-cocycle relations

g αγ=g βγg αβ:U αU βU γGL( n)g_{\alpha \gamma} = g_{\beta\gamma} \circ g_{\alpha\beta}: U_{\alpha} \cap U_\beta \cap U_\gamma \to GL(\mathbb{R}^n)
g αα=1:U αGL( n)\qquad g_{\alpha\alpha} = 1: U_{\alpha} \to GL(\mathbb{R}^n)

In general, given any collection of transition maps that satisfy these cocycle conditions, we can construct a vector bundle in the same manner (which may be different from the tangent bundle).

Definition in synthetic differential geometry

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 (𝒯,(R,+,))(\mathcal{T},(R,+,\cdot)) be a smooth topos and write D={ϵR|ϵ 2=0}D = \{\epsilon \in R| \epsilon^2 = 0\} for the standard infinitesimal interval. For X𝒯X \in \mathcal{T} any object (any space in 𝒯\mathcal{T}), the tangent bundle of XX is the morphism

p:TXX p : T X \to X


  • TX∶−X DT X \coloneq X^D the internal hom of DD into XX;

  • p=ev 0p = ev_0 the evaluation map at the origin of DD

    ev 0:(UvX D)(U×*Id×0U×Dv¯X)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 v¯\bar v is the hom-adjunct of vv.

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 DX^D are fiberwise linear, i.e. are fiberwise RR-modules the way one expects. Objects XX for which this is true are microlinear spaces in 𝒯\mathcal{T}. See there for more details.

A smooth topos 𝒯\mathcal{T} is called a well-adapted model for synthetic differential geometry if there is a full and faithful embedding Diff 𝒯\hookrightarrow \mathcal{T} of the cageory of manifolds into 𝒯\mathcal{T}.

Typically, for well adapted models, under this embedding

  • manifolds are microlinear spaces

  • the synthetic definition of tangent bundle X DX^D for XX a manifold does coincide with the ordinary notion of TXT X.

Let 𝕃=(C Ring fin) op\mathbb{L} = (C^\infty Ring^{fin})^{op} be the category of smooth loci. For MM a manifold, the exponential M DM^D does exist in 𝕃\mathbb{L} and is isomorphic to the ordinary tangent bundle TXT X of XX. (For instance MSIA, chapter II, prop 1.12.

There are well-adapted smooth toposes 𝒵\mathcal{Z} and \mathcal{B} presented as categories of sheaves on 𝕃\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:𝕃Sh(𝕃)Y : \mathbb{L} \to Sh(\mathbb{L}) does preserve the above property.

Hence in these models for XDiffX \in Diff a manifold, TXDiffT X \in Diff its ordinary tangent bundle and s:DiffSh(𝕃)s : Diff \to Sh(\mathbb{L}) the full and faithful embedding, we have isomorphisms

(s(X)) Ds(TX) (s(X))^D \simeq s(T X)

which respect the bundle maps.

As a supermanifold

The tangent bundle of a manifold XX may be interpreted as a supermanifold in which XX has degree 00 and the tangent vectors have degree 11. See shifted tangent bundle.

Definition for other generalized smooth spaces

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

geometrypointfirst order infinitesimal\subsetformal = arbitrary order infinitesimal\subsetlocal = stalkwise\subsetfinite
\leftarrow differentiationintegration \to
smooth functionsderivativeTaylor seriesgermsmooth function
curve (path)tangent vectorjetgerm of curvecurve
smooth spaceinfinitesimal neighbourhoodformal neighbourhoodgerm of a spaceopen neighbourhood
function algebrasquare-0 ring extensionnilpotent ring extension/formal completionring extension
arithmetic geometry𝔽 p\mathbb{F}_p finite field p\mathbb{Z}_p p-adic integers (p)\mathbb{Z}_{(p)} localization at (p)\mathbb{Z} integers
Lie theoryLie algebraformal grouplocal Lie groupLie group
symplectic geometryPoisson manifoldformal deformation quantizationlocal strict deformation quantizationstrict 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

Revised on February 7, 2017 11:00:45 by Urs Schreiber (