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• W2043923114 abstract "In this note it is proven that a regular Riemannian s-manifold of noncompact type (see below) cannot be immersed isometrically and equivariantly in R. Our notation, terminology and basic facts will be those of [3]. Let (M, {S,}) be a connected periodic regular s-manifold which is metrizable, i.e. there is a Riemannian metric g on M which is invariant with respect to the symmetries {S,: x E M}. (Periodicity means that (M, {S,)) has finite order [3, p. 4].) We have the group of isometries I(M, g) which is transitive on M [3, p. 2]. Contained in I(M, g) we have the group of transvections G = Tr(M, {S }) [3, p. 57] which is generated by the elementary transvections, i.e. by the isometries Sx o Sy-', x, y E M. About the group G one knows: (1) G is a connected Lie group [3, II 32, 125]. (2) G is transitive on M [3, II 33]. It is known [3, IV 24] that under the above conditions (M, {S5}) admits two complementary foliations F, 2 such that: (a) IF is invariant and its leaves are regular s-manifolds with solvable group of transvections. (b) 2 is weakly invariant and its leaves are regular s-manifolds with semisimple group of transvections (compare [2, p. 208]). DEFINITION. We shall say that (M, {S,}) is of noncompact type if the foliation S2 has noncompact leaves. The objective of this note is to prove the following. THEOREM. Let (M, {S,1) be a connected periodic, regular s-manifold which is metrizable and of noncompact type. Then (M, {S,}) admits no isometric equivariant immersion into a finite-dimensional real representation of G = Tr(M, {S,}). PROOF. Let us assume the existence of such an isometric immersion (p, f): (G, M) -(I(Rn), Rn), where T is a Lie group monomorphism and f is an isometric Received by the editors January 8, 1982. AMS (MOS) subject classifications (1970). Primary 53C40, 53B25." @default.
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• W2043923114 date "1983-01-01" @default.
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• W2043923114 title "Regular Riemannian $s$-manifolds of noncompact type" @default.
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