nLab generalized tangent bundle

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

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

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

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 }

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Definition

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.

Properties

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
Spin(7)-structureSpin(8)Spin(7)
G2-structureSpin(7)G2
CY3-structureSpin(6)SU(3)
SU(2)-structureSpin(5)SU(2)
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.