Assume that $M$ is a compact parallelizable manifold. Is there an upper bound for the absolute value of Fredholm index of all operators in the form $D=\sum_{i=1}^n \partial^2/\partial{X_i^2}$ where $\{X_1,X_2,\ldots,X_n\}$ is a global smooth frame?
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1$\begingroup$ $M$ is connected I guess? $\endgroup$ – Thomas Rot Jun 21 at 3:51

$\begingroup$ @ThomasRot Yes we assume that $M$ is connected. But even if $M$ is not connected it can not have infinite number of connected component. $\endgroup$ – Ali Taghavi Jun 21 at 5:14

1$\begingroup$ Search of "quasi Laplace operator" in Google brings "Laplace quasioperator". $\endgroup$ – user64494 Jun 21 at 8:28

$\begingroup$ @user64494 thanks for your edit. $\endgroup$ – Ali Taghavi Jun 21 at 11:08