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Let's say we have a vector bundle $V$ on a compact Riemannian manifold $M$ (of dimension $m$, with metric $g$). Given a differential operator $P$, acting on the sections of $V$, of the form: $$P = \nabla^* \nabla + E,$$ where $\nabla$ is a connection on $V$, with $\nabla^* \nabla$ being the connection Laplacian, and $E$ is an endomorphism of $V$.

Is it necessary for $E$ to be invariant under a diffeomorphism $\phi : M \rightarrow M$?

In particular, under general covariance (invariance of the form under arbitrary differentiable coordinate transformations)?

For more context see:

  1. Section 11.2 of http://www.alainconnes.org/docs/bookwebfinal.pdf
  2. Lemma 4.8.1 of https://pages.uoregon.edu/gilkey/dirPDF/InvarianceTheory1Ed.pdf
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    $\begingroup$ Necessary for what? There could be a lot of diffeomorphism of $M$. If you have no conditions the operator should satisfy, there are no constraints on $E$ from $\phi$. $\endgroup$
    – quarague
    Commented Feb 7, 2019 at 9:23
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    $\begingroup$ You also need a metric on $V$ to define $\nabla^*$. Also, a diffeomorphism of $M$ need not induce an action on $V$. $\endgroup$
    – Liviu Nicolaescu
    Commented Feb 7, 2019 at 9:41
  • $\begingroup$ The wikipedia article makes the case that the term "general covariance" is very unclear. Can you perhaps express what you want from "general covariance" in more precise terms, saying what should be invariant under what? Do you only want invariant equations of motion, or do you want invariant Lagrangians, or invariant integrals of those Lagrangians? And invariant under what sort of transformations? $\endgroup$
    – Ben McKay
    Commented Feb 7, 2019 at 10:54

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