The tangent bundle of a (sufficiently differentiable) space is a bundle over whose fiber over a point is the tangent space at that point, namely the collection of infinitesimal curves in emanating at : “tangent vectors”.
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 at is an element of .
The tangent space of at a point is the fiber of over ;.
A tangent vector field on is a section of .
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.
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 at on as a scalar-valued derivation on the space of germs of differentiable functions defined on near , augmented by evaluation at . That is, given partial functions and , each defined on some neighbourhood of , we have:
In light of (4), (3) is equivalent to:
Globally, we may define a tangent vector field as an ordinary (unaugmented) derivation on the space of differentiable functions defined on all of . (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 and , we have:
In light of (4), (3) is again equivalent to:
Given a differentiable curve , the derivative of is a curve in the tangent bundle; given an argument and a function defined near , the action is given by
where indicates the usual derivative on the real line, so that is a tangent vector at . (We really only need to be defined on a neighbourhood of , of course.)
(Of course, could be replaced by any argument in this definition.)
A particularly important case is when is a level curve in some system of local coordinates at ; that is, is the point whose th coordinate is and whose other coordinates are the same as at . The local tangent vector field given by these curves may be written or (note the placement of the scripts). This is because, if a function defined on that coordinate patch is identified with a function of real variables, then becomes identified with the partial derivative . In general, a local vector field on such a coordinate patch can be written
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 doesn't make sense for an arbitrary function ; it only makes sense when is given as one component of a coordinate system. That is, if and are both coordinate systems, then the two meanings of need not be the same, because constant and constant aren't the same condition. However, we can use the more complicated notation or ; this is common when is a phase space in thermodynamics. Of course, if a coordinate system is fixed by convention, then there is no ambiguity.
If is a smooth -manifold, then we know the tangent space at each point is isomorphic to . We can exploit this fact to construct the tangent space by gluing many copies of together. However, one drawback is that it is not immediately obvious that the result is independent of the atlas chosen.
Let be a smooth -manifold defined by an atlas . Then we may define its tangent bundle by a gluing construction in , taking to be the quotient of the disjoint sum
obtained by dividing by the equivalence relation
where , and is the result of differentiating the transition function at the point . We thus obtain a covering of , and these form coordinate charts of a smooth manifold structure on in a more or less evident way, and the projection map is given by
In the above construction, the charts were not used directly; Instead, the transition maps
are what were needed to construct the tangent bundle. These maps satisfy Čech 1-cocycle relations
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).
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)
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 are fiberwise linear, i.e. are fiberwise -modules the way one expects. Objects for which this is true are microlinear spaces in . See there for more details.
Typically, for well adapted models, under this embedding
manifolds are microlinear spaces
the synthetic definition of tangent bundle for a manifold does coincide with the ordinary notion of .
There are well-adapted smooth toposes and presented as categories of sheaves on : 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 does preserve the above property.
Hence in these models for a manifold, its ordinary tangent bundle and the full and faithful embedding, we have isomorphisms
which respect the bundle maps.
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.
Examples of sequences of local structures
|geometry||point||first order infinitesimal||formal = arbitrary order infinitesimal||local = stalkwise||finite|
|smooth functions||derivative||Taylor series||germ||smooth function|
|curve (path)||tangent vector||jet||germ of curve||curve|
|smooth space||infinitesimal neighbourhood||formal neighbourhood||germ of a space||open neighbourhood|
|function algebra||square-0 ring extension||nilpotent ring extension/formal completion||ring extension|
|arithmetic geometry||finite field||p-adic integers||localization at (p)||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