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Let $D$ be an elliptic differential operator defined on $\Gamma(E)$ where $\Gamma (E)$ is the space of smooth sections of vector bundle $E$ over a smooth manifold $M$. So $\Gamma (E)$ is a $C^\infty(M)$-module where $C^\infty (M)$ is the space of all smooth functions on $M$.

Question: Is it true to say that the module generated by $\ker D$ is the whole $\Gamma (E)$?

Motvation: Motivation comes from the method of variation of parameters in Differential equation. Recall that variation of parameters is based on the following method: we have a second order diff. eq. $$y''+P(x)y'+Q(x)y=g(x)$$(which is realy an elliptic equation). So we assume that a general solution is in the form $$v_1y_1+v_2y_2$$ where $y_1,y_2$ are solutions of the corresponding homogenious equation $$D(y)=y''+P(x)y'+Q(x)y=0$$ and $v_1,v_2$ are unknown functions. So for every arbitrary function $h$ there exist a $y$ in the form $y=v_1y_1+v_2y_2$ such that $g:=D(h)=D(y)$. This means that every arbitrary function $h$ belongs to the module generated by $\ker D$

Is there a terminology for those differential operators $D$ for which $\Gamma(E)$ is equal to the $C^\infty$-module generated by $\ker D$?

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    $\begingroup$ Take the laplacian on a compact manifold $X$ acting on the space of $p$-forms: the kernel is isomorphic to $H^p(X,\mathbb{R})$, which may very well be zero. $\endgroup$
    – abx
    Commented Jul 17, 2022 at 8:54
  • $\begingroup$ @abx Thank you for your interesting answer. What about a possible counter example in the case of open manifold? $\endgroup$
    – Ali Taghavi
    Commented Jul 17, 2022 at 9:00
  • $\begingroup$ @AliTaghavi For a non-compact example take $X \times \mathbb{R}^k$ for any $k>0$. $\endgroup$
    – Igor Khavkine
    Commented Jul 17, 2022 at 9:02
  • $\begingroup$ @IgorKhavkine I assume you mean $k>1$ $\endgroup$
    – Ali Taghavi
    Commented Jul 17, 2022 at 9:10
  • $\begingroup$ @abx I wonder if there is a terminology for those operator $D$ with this property? $\endgroup$
    – Ali Taghavi
    Commented Jul 17, 2022 at 9:11

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