## Adiabatic Limit and the Bott Connection

Background
Let $M$ be an even dimensional oriented closed spin manifold. We define the $\hat{A}$-genus of $M$, denote by $\hat{A}(M)$, by
$\hat{A}(M)=\big\langle \hat{A}(TM),[M] \big\rangle= \int_M\hat{A}(TM,\nabla^{TM})\in\Z.$ Continue Reading

## Foliations and the Bott Vanishing Theorem

Let $M$ be a closed manifold and $TM$ its tangent vector bundle. Let $F\subset TM$ be a sub-vector bundle of $TM$. Define
$F ~\text{is integrable} ~\Leftrightarrow ~~\Big(\forall X,Y\in \Gamma(F) \Big)~ [X,Y]\in \Gamma(F),$
where $[\cdot,\cdot]$ is Lie bracket. Note the operator Lie bracket is only defined on tangent bundle. Continue Reading

## The Chern-Simons Transgressed Form

In the end of the proof of theorem1.9, we have
$tr \left[ f(R^E)\right]-tr \left[ f(\tilde{R}^E)\right]=-d\int_0^1tr \left[ \frac{d\nabla_t^E}{dt}f'(R_t^E)\right]dt.$
The transgressed term
$-d\int_0^1tr \left[ \frac{d\nabla_t^E}{dt}f'(R_t^E)\right]dt$ Continue Reading

## K-groups and the Chern Character

Let $E$ be a complex vector bundle over a compact smooth manifold $M$. Let $\nabla^E$ be a (C-linear) connection on $E$ and let $R^E$ denote its curvature.

The Chern character form associated to $\nabla^E$ is defined by
$ch\left(E,\nabla^E\right)=tr\left[ \exp\left(\frac{\sqrt{-1}}{2\pi}R^E\right)\right]\in\Omega^{even}(M).$ Continue Reading