nLab
Bott connection

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

tangent cohesion

differential cohesion

graded differential 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}{}& ʃ &\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)

Higher Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

In foliation-theory the Bott connection is a certain characteristic canonucal on the conormal bundle of the foliation.

Definition

Definition

For XX a smooth manifold and 𝒫TX\mathcal{P} \hookrightarrow T X a foliation of XX, incarnated as a subbundle of the tangent bundle, the corresponding conormal bundle

𝒫 T *X \mathcal{P}^\perp \hookrightarrow T^* X

is the annihilator of 𝒫\mathcal{P} under the pairing of covectors with vectors. The corresponding Bott connection is the covariant derivative of vectors XΓ(𝒫)X \in \Gamma(\mathcal{P}) on covectors ξΓ(𝒫 )\xi \in \Gamma(\mathcal{P}^\perp) given by the Lie derivative

X:ξ Xξ=ι Xd dRξ. \nabla_X \;\colon\; \xi \mapsto \mathcal{L}_X \xi = \iota_X d_{dR} \xi \,.

Similarly there is a Bott connection along 𝒫\mathcal{P} along the normal bundle TX/𝒫T X / \mathcal{P}. More generally, one speaks of a Bott connection for any connection on the (co)normal bundle and defined on all vector fields on XX which restricts to the above along leaves.

References

The notion originates in

  • Raoul Bott, Lectures on characteristic classes and foliations, in Lectures on algebraic and differential topology, Springer Lecture Notes in Mathematics, 279, (1972)

in the study of characteristic classes of foliations. Further developments are in

  • Jonathan Bowden, On foliated characteristic classes of transversally symplectic foliations (arXiv:1108.1919)

Created on March 25, 2013 at 20:36:06. See the history of this page for a list of all contributions to it.