Consider $\mathbb{R}^{n+1}$ equipped with the Minkowski (sign indefinite) metric:
$$g=(x^0)^2-(x^1)^2-\dots -(x^n)^2.$$



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> Is there a classification of diffeomorphisms $F\colon \mathbb{R}^{n+1}\tilde\to \mathbb{R}^{n+1}$ with the property $F^\*g=a\cdot g$, where $a$ is a constant?
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Classification of similarity transformations of Minkowski space
If $a>0$, since $a$ is constant, you can just compose $F$ with a suitable rescaling to get $a=1$, and then $F$ is an isometry. So isometries composed with dilations. If $a<0$, not possible for $n>1$ because it changes signature.


