Let us consider the so-called Lichnerowicz equation

on , a Riemannian manifold of dimension . Here , , and are smooth function with .

- We first use the the following variable change
Clearly,

and

Therefore, we can write

Using this rule, we can rewrite the equation as follows

Clearly, under this variable change, we have killed after in the original equation.

- We now use the change
Clearly,

and

Therefore, we can write

Again, our PDE becomes

This seems that this variable change is not useful and is useful only when we deal with PDEs with nonlinearities of power type.

- Finally, we use the change
Clearly,

and

Therefore, we can write

Again, our PDE becomes

Depending on each problem, we may choose a suitable variable change.

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