nLab Bott connection

Contents

Context

Differential geometry

synthetic differential geometry

Introductions

from point-set topology to differentiable manifolds

Differentials

V-manifolds

smooth space

Tangency

The magic algebraic facts

Theorems

Axiomatics

cohesion

tangent cohesion

differential cohesion

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

Models

Lie theory, ∞-Lie theory

differential equations, variational calculus

Chern-Weil theory, ∞-Chern-Weil theory

Cartan geometry (super, higher)

Contents

Idea

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

Definition

Definition

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

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

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

$\nabla_X \;\colon\; \xi \mapsto \mathcal{L}_X \xi = \iota_X d_{dR} \xi \,.$

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

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)