Newsgroups: sci.math.research From: mblau@enslapp.ens-lyon.fr (Matthias Blau) Subject: Re: Is there an equivariant isometric embedding theorem? Date: Fri, 3 Mar 1995 10:30:54 GMT In article <3j58s4$hf2@fidoii.cc.lehigh.edu>, dlj0@lehigh.edu (DAVID L. JOHNSON) asks: |> If M is a compact Riemannian manifold, and if G acts on M via isometries, is |> there an N >> 0 and an isometric embedding of M into R^N so that the action |> of G becomes orthogonal? That is, is there an equivariant version of Nash's |> theorem? |> |> I know there are equvariant embedding theorems, but I don't know whether |> they have been extended to isometric embeddings. This is certainly true for homogeneous spaces, according to a theorem by Moore and Schlafly (Math Z 173 (1980) 119), i.e. one has the following theorem: Theorem: Let G be a compact Lie group and G/H a compact homogeneous space with a G-invariant Riemannian structure. Then there is an isometric embedding G/H --> R^{m} for some m and a monomorphism G --> O(m) such that ... (the two commute in the obvious way). I don't know if the equivariant embedding theorems of Palais and Mostow have been generalized to isometric embeddings for other manifolds, but I believe (can't check this here, sorry) that the theorem of Moore and Schlafly is more general than the version I gave above (which I found in the book "Geometry of non-linear field theories" by Percacci). Regards, matthias -- Matthias Blau Laboratoire de Physique Theorique Ecole Normale Superieure de Lyon mblau@enslapp.ens-lyon.fr ============================================================================== From: "Roger Schlafly" To: Subject: diff. geo. links Date: Sat, 25 Jul 1998 16:00:55 -0700 [deletia] I would just like to add one comment. > theorem of Moore and Schlafly is more general than the version I gave above Yes, indeed it is. The above theorem was first proved by Moore in a separate paper. In the above paper, Moore and I proved that * compact riemannian manifolds always have equivariant isometric embeddings into R^n. * there are no reasonable bounds on n. * if the manifold is not compact (but the group is compact), it may not embed in any R^n, but will embed in hilbert space. * pseudo-riemannian manifolds embed in some R^n with indefinite metric Roger Schlafly