# dg.differential geometry: a cohomology associated with a vector field in a variety of Riemann

Leave $$X$$ be a vector field in a Riemannian variety $$(M, g)$$. So we have a $$1$$-to form $$beta$$ with $$beta (Y) =_g$$.

We consider the following subcomplex of the Rham complex $$Omega ^ * (M)$$: $${ alpha en Omega ^ * (M) | d alpha = alpha wedge beta }$$

This generates a cohomology with the standard external derivation. For the zero vector field we obtain the standard Rham cohomology.

Is this cohomology always a finite dimensional space? Does it depend on the Riemannian structure? Does it contain any dynamic information about the vector field? $$X$$?

Is there an appropriate analogy of this cohomology in algebraic topology or another type of cohomologist? (In the context of algebraic topology, we replace $$1$$-to form $$beta$$ with a $$1$$-cochain and the outer derivative with cup product)