The exterior derivative $d$ has many very nice algebraic relations. For example

1. $d(\alpha\wedge\beta) = (d\alpha)\wedge \beta + (-1)^k\alpha\wedge(d\beta)$
2. $f^*(d \alpha)=d f^*(\alpha)$.
3. $d\circ d = 0$

for $\alpha, \beta$ forms on a manifold $V$ and $f:V\rightarrow W$ a smooth application.

Let $\delta = \star d \star$ the codifferential, we have $\delta\circ \delta =0$. I wonder if there are other simple and usefull properties as above. For instance can we say anything about $\delta f^*(\alpha)$?

The exterior derivative $d$ has many very nice algebraic relations. For example

1. $d(\alpha\wedge\beta) = (d\alpha)\wedge \beta + (-1)^k\alpha\wedge(d\beta)$
2. $f^*(d \alpha)=d f^*(\alpha)$.
3. $d\circ d = 0$

for $\alpha, \beta$ forms on a manifold $V$ and $f:V\rightarrow W$ a smooth map.

Let $\delta = \star d \star$ the codifferential, we have $\delta\circ \delta =0$. I wonder if there are other simple and usefull properties as above. For instance can we say anything about $\delta f^*(\alpha)$?