I am currently studying a French paper on Einstein manifolds by Berard Bergery and I have doubts that my translation of the following sentence is correct:

De plus, puisque $G$ agit par isometries, $M/G$ herite par quotient d'une metrique, qui est enfait une metrique riemannienne ("a bord" eventuellement).

My translation:

Furthermore, since $G$ acts by isometries, the quotient $M/G$ inherits a metric, which is actually a Riemannian metric (possibly (..)).

Is this correct so far? I don't know how to translate the last two words, apologies for leaving out the accents!

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    $\begingroup$ "Une variété à bord" translates as "a manifold with boundary", so here we might have "a riemannian metric, possibly with boundary". But I don't know if such a concept exists (the standard books on differential manifolds don't seem to cover riemannian metrics on manifolds with boundary) $\endgroup$ – Georges Elencwajg Sep 10 '13 at 14:17
  • $\begingroup$ "À bord" is "with boundary". Ref. $\endgroup$ – Did Sep 10 '13 at 14:19
  • $\begingroup$ @Did: les bons esprits se rencontrent :-) $\endgroup$ – Georges Elencwajg Sep 10 '13 at 14:20
  • $\begingroup$ @GeorgesElencwajg En vérité. Mais votre commentaire (que je n'avais pas vu en tapant le mien) est plus complet. $\endgroup$ – Did Sep 10 '13 at 14:22
  • $\begingroup$ @GeorgesElencwajg this is what confused me, too (i.e. a Riemannian metric with boundary). $\endgroup$ – harlekin Sep 10 '13 at 14:23

Furthermore, since $G$ acts by isometries, $M/G$ inherits by the quotient operation a metric, which is in fact a Riemannian metric (possibly with boundary).

Probably the last part means "the metric of a Riemannian manifold, possibly with boundary", but the French is not quite saying that.

Note that "enfait" should have been "en fait".

  • $\begingroup$ Yes, indeed that should have been "en fait" (I'll leave it as it is so as to not interfere indirectly with your answer). Many thanks! $\endgroup$ – harlekin Sep 10 '13 at 14:25

Good except "$M/G$ hérite par quotient d'une métrique" actually means "$M/G$ inherits a metric by quotient".

"à bord eventuellement" means "possibly on the edge" but not sure of the context.

I guess you got the meaning right.

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    $\begingroup$ "On the edge": no. $\endgroup$ – Did Sep 10 '13 at 14:20

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