nLab generalized tangent bundle



Differential geometry

synthetic differential geometry


from point-set topology to differentiable manifolds

geometry of physics: coordinate systems, smooth spaces, manifolds, smooth homotopy types, supergeometry



smooth space


The magic algebraic facts




infinitesimal cohesion

tangent cohesion

differential cohesion

graded differential cohesion

singular cohesion

id id fermionic bosonic bosonic Rh rheonomic reduced infinitesimal infinitesimal & étale cohesive ʃ discrete discrete continuous * \array{ && id &\dashv& id \\ && \vee && \vee \\ &\stackrel{fermionic}{}& \rightrightarrows &\dashv& \rightsquigarrow & \stackrel{bosonic}{} \\ && \bot && \bot \\ &\stackrel{bosonic}{} & \rightsquigarrow &\dashv& \mathrm{R}\!\!\mathrm{h} & \stackrel{rheonomic}{} \\ && \vee && \vee \\ &\stackrel{reduced}{} & \Re &\dashv& \Im & \stackrel{infinitesimal}{} \\ && \bot && \bot \\ &\stackrel{infinitesimal}{}& \Im &\dashv& \& & \stackrel{\text{étale}}{} \\ && \vee && \vee \\ &\stackrel{cohesive}{}& \esh &\dashv& \flat & \stackrel{discrete}{} \\ && \bot && \bot \\ &\stackrel{discrete}{}& \flat &\dashv& \sharp & \stackrel{continuous}{} \\ && \vee && \vee \\ && \emptyset &\dashv& \ast }


Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)



In the context of generalized complex geometry one says for XX a manifold, TXT X its tangent bundle and T *XT^* X the cotangent bundle that the fiberwise direct sum-bundle TXT *XT X \oplus T^* X is the generalized tangent bundle.

More generally, a vector bundle EXE \to X that sits in an exact sequence T *XETXT^* X \to E \to T X is called a generalized tangent bundle, such as notably those underlying a Courant Lie 2-algebroid over XX.


As an associated bundle

The ordinary tangent bundle is the canonical associated bundle to the general linear group-principal bundle classified by the morphism

g TX:XBGL(n) g_{T X} : X \to \mathbf{B} GL(n)

to the smooth moduli stack of GL(n)GL(n).

Similarly there is a canonical morphism

(g TX,g TX T):XBO(n,n) (g_{T X}, g^{-T}_{T X}) : X \to \mathbf{B} O(n,n)

to the moduli stack which is the delooping of the Narain group O(n,n)O(n,n). This classifies the O(n,n)O(n,n)-principal bundle to which TXT *XT X \oplus T^* X is associated.

Reduction of structure group

Where a reduction of the structure group of the tangent bundle along BO(n)BGL(n)\mathbf{B} O(n) \hookrightarrow \mathbf{B} GL(n) is equivalently a vielbein/orthogonal structure/Riemannian metric on XX, so a reduction of the structure group of the generalized tangent bundle along B(O(n)×O(n))BO(n,n)\mathbf{B} (O(n) \times O(n)) \to \mathbf{B}O(n,n) is a generalized vielbein, defining a type II geometry on XX.

Other reductions yield other geometric notions, for instance:

Spin(8)-subgroups and reductions to exceptional geometry

reductionfrom spin groupto maximal subgroup
generalized reductionfrom Narain groupto direct product group
generalized Spin(7)-structureSpin(8,8)Spin(8,8)Spin(7)×Spin(7)Spin(7) \times Spin(7)
generalized G2-structureSpin(7,7)Spin(7,7)G 2×G 2G_2 \times G_2
generalized CY3Spin(6,6)Spin(6,6)SU(3)×SU(3)SU(3) \times SU(3)

see also: coset space structure on n-spheres

Last revised on March 30, 2019 at 14:01:06. See the history of this page for a list of all contributions to it.