Abstract
In this short note, we present certain generalized versions of the commutator formulas of some natural operators on manifolds, and give some applications. The purpose of this note is to present several general commutator formulas of certain natural operators on Riemannian manifolds, complex manifolds and generalized complex manifolds. We would like to point out that such commutator formulas are essentially consequences of the classical Cartan formula for Lie derivative, but they have deep applications in geometry such as in studying the smoothness of deformation spaces of manifolds. For example one direct consequence of the commutator formula is the Tian-Todorov lemma which is essential for proving the smoothness of the deformation space of Calabi- Yau manifolds in (10, 11) and also (12). The general commutator formulas derived in the note also have applications in proving smoothness of more general deformation spaces such as that of the generalized complex manifolds in (5). We will discuss the applications of these commutator formulas in deformation theory in our subsequent work.
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