At the beginning a word of warning: this would be rather vague question: vague as it is, I'm not requiring a precise answer, rather some intuitive explanation.

In the flat case $M=\mathbb{R}^n$ there are some naturally constructed differential operators: the Laplace operator, the Hodge de-Rham operator $d+d^*$, the signature operator $d+d^*$ (using different grading), Dirac operator etc.

In the general case of arbitrary manifolds Dirac operator does not always exist: the manifold should be $spin^{c}$. On the contrary, for the Laplace operator to make sense no assumption on $M$ is required.

Suppose that I have some differential operator on $\mathbb{R}^n$: is there some way to answer immediately whether this operator makes sense on general manifolds or there will be some topological constraints for defining it on arbitrary manifold?