# A uniform upper bound for Fredholm index of Laplace quasi-operators on a compact parallelizable manifold

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?

• $M$ is connected I guess? – Thomas Rot Jun 21 at 3:51
• @ThomasRot Yes we assume that $M$ is connected. But even if $M$ is not connected it can not have infinite number of connected component. – Ali Taghavi Jun 21 at 5:14
• Search of "quasi Laplace operator" in Google brings "Laplace quasi-operator". – user64494 Jun 21 at 8:28
• @user64494 thanks for your edit. – Ali Taghavi Jun 21 at 11:08