Motivated by classical formulas $L_{X}=d\circ i_{X}+i_{X}\circ d$ and $L_{X} \Omega=Div(X) \Omega$ and the essential role of the diff operator $d$ in definition of divergence, we define some quantities, say $\overline{Div}$, based on the adjoint operator $d^{*}=\pm *d*$, where $*$ is the Hodge star operator.
In this way our main question is that:
What are some geometric or physical interpretations for $\overline{Div}$? What are some calculus identities for this quantity?In particular is it true that for a closed manifold $M$, with volum form $\Omega$, we have $\int_{M} \overline{Div}(X)\Omega=0$?
Moreover what is the dynamical interpretation of $\overline{Div}(X)=0$. This is motivated by classical case: If $Div(X)=0$ then $X$ has no an attractor, since the flow of $X$ generates a one parameter family of volume preserving diffeomorphisms. So we ask: Is there a vector field $X$ which has a compact attractor invariant set but $\overline{Div}(X)$ is identically zero?
- For a vector field $X$ on a $2$ dimensional surface with volum form $\Omega$ define:
$$\overline{Div}(X)=(i_{X}\circ d^{*}+d^{*}\circ i_{X})(\Omega)$$
A vector field $X$ on a Riemannian manifold $(M,g)$ defines a one form $\alpha$. Now $\overline{Div}(X)$ is defined as a unique function with $$\alpha \wedge d^{*}(\Omega)=\overline{Div}(X). \Omega $$
For a symplectic manifold $(M,\omega)$, $\overline{Div}$ is the unique function with $$ (i_{X}\circ d^{*}+d^{*}\circ i_{X})(\Omega)\wedge \omega=\overline{Div}(X).\Omega$$ where $\Omega$ is the corresponding volume form.