2
$\begingroup$

Let $X$ be a complete Riemannian manifold and $H$ be the kernel of generalized Dirac operator $D$ on $L(S)$, where $S$ is the Dirac bundle. Let $K$ be a compact subset of $X$ and $K\subset \Omega$ be a neighborhood of $K$. Fix an $\varepsilon>0$ and choose an $\varepsilon$-dense subset $\{x_m\}_{m=1}^d$ of $K$, i.e., every point of $K$ is within distance $\varepsilon$ of some $x_m$.

My question: If we suppose $\dim H>d$, why there exists an element $\sigma\in H$ such that $\|\sigma\|=1$ and $\sigma(x_m)=0$ for all $m=1,\cdots,d$?

Could you please help me with the details? Thank you!

Improve this question
$\endgroup$
4
  • $\begingroup$ It appears that you are reading some paper about solutions of a generalized Dirac equation. If you want help understanding the paper, you should at least tell us which paper you are reading. At the very least, it will help us understand the notation you are using. (For example, what do you mean by $L(S)$ and what do you mean by $\| \sigma\|$?) $\endgroup$
    – Robert Bryant
    Commented Oct 25, 2021 at 21:26
  • $\begingroup$ @RobertBryant Yeap，it is the paper of Gromov Lawson in 1983. $\endgroup$
    – Radeha Longa
    Commented Oct 26, 2021 at 9:44
  • $\begingroup$ @RobertBryant $\|\sigma\|=\int_X <\sigma_1,\sigma_2>$, where $\sigma_1,\sigma_2$ are compactly supported in the space $\Gamma(S)$ of smooth of Dirac bundle $S$. And $L(S)$ is the completion of $\Gamma(S)$ with respect to the norm $\|\cdot\|$. Could you give me some help? Beside this question, I'm also confused with a theorem, this is the linking mathoverflow.net/questions/407049/… $\endgroup$
    – Radeha Longa
    Commented Oct 26, 2021 at 12:40
  • $\begingroup$ numdam.org/item/PMIHES_1983__58__83_0.pdf $\endgroup$
    – Chris Gerig
    Commented Oct 27, 2021 at 22:23

0

Reset to default

You must log in to answer this question.