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

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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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.

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  • $\begingroup$ Thanks. Is there an explicit classification of isometries? Are they necessarily affine? $\endgroup$
    – asv
    Sep 8, 2020 at 14:47
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    $\begingroup$ Isometries preserve the Levi-Civita connection, which is the standard flaf connection of Euclidean space, and preserving that is precisely being affine. So then you see that the isometries are Poincare transformations, i.e. translations and Lorentz transformations. $\endgroup$
    – Ben McKay
    Sep 8, 2020 at 14:49

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