Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

Let $M$ be an n-dimensional closed oriented Riemannian manifold.

For any $e\in\Gamma(TM)$, let $e^{*}\in\Gamma(T^{\ast}M)$ corresponds to e via $g^{TM}$.

Let $\hat{c}(e)$ be the Clifford operator acting on the exterior algebra bundle $\wedge^{*}(T^{\ast}M)\otimes\mathbb{C}$ defined by

$$\hat{c}(e)=e^{*}\wedge+i_{e}$$

where $e^{*}\wedge$ and $i_{e}$ are the standard notation for exterior and interior multiplications.

We known there is a de Rham-Hodge operator $$D=d+d^{*}.$$

The usually deformation of de Rham-Hodge operator by a vector field $V\in\Gamma(TM)$ is

$$D_{T}=d+d^{*}+T\hat{c}(V)$$

I want to discuss the deformation of de Rham-Hodge operator by two vector field

$X,Y\in \Gamma(TM)$, defined by

$$D_{T}=d+d^{*}+T(\hat{c}(X)+\sqrt{-1}\hat{c}(Y))$$ for any $T\in\mathbb{R}$.

We known $$D_{T}^{2}=D^{2}+TD(\hat{c}(X)+\sqrt{-1}\hat{c}(Y))+T^{2}(\hat{c}(X)+\sqrt{-1}\hat{c}(Y))(\hat{c}(X)+\sqrt{-1}\hat{c}(Y))$$

I have two questionss:

1、Is the $D_{T}^{2}$ a $Schr\ddot{o}dinger$ type operator?

2、How to discuss the operator $D_{T}^{2}$ in "local"? I mean in small neighborhoods of zeros of $(\hat{c}(X)+\sqrt{-1}\hat{c}(Y))(\hat{c}(X)+\sqrt{-1}\hat{c}(Y))$.

Remark:The notations come form the book 'Lectrues on Chern-Weil theory and Witten deformations' by Weiping Zhang.

share|improve this question
add comment

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Browse other questions tagged or ask your own question.