proofs of Theorems \[heb-vaug\] and \[main2\]: Roughly speaking, Theorem \[heb-vaug\] uses local charts compatible with the dimension reduction under the Riemannian submersion induced by the isometries, reducing the compact embedding of functions to a Sobolev inequality in the space orthogonal to the $H$-orbits, providing a convenient setup for specific results obtained by H. Berestycki, E. Lieb, P. L. Lions and others, see [@lions1]. The proof of Theorem \[main2\] is different: the dimension reduction compatible with isometries used in Theorem \[heb-vaug\] is not always compatible with volume preserving diffeomorphisms.[^6] Some ingredients in the proof are a Sobolev-Hajłasz inequality [@Hajlasz], and variations of the Concentration-Compactness principle, see [@lions2].\

In Section \[Preliminaries\] we provide definitions and results that will be used in Section \[main\], where a detailed proof of Theorem \[main2\] is given. In Section \[examples\] we see that Theorem \[main2\] can be applied in the Riemannian context, and discuss necessary and sufficient conditions for its applicability when the dimension of the $H$-orbits is 1.\
For Sobolev embeddings in non-compact spaces using symmetry, see [@Gaczkowski; @gorka1], and the references there.

Preliminaries {#Preliminaries}
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In this work $(X, d, \mu)$ is a metric-measure space equipped with a metric $d$ and a Radon measure $\mu$. 