Equivariant embeddings of manifolds into Euclidean spaces

2022/08/31 by Zhongzi Wang, Wang, Zhongzi · 2 citations
Mathematics · #20C30 #20H10 #Advanced Operator Algebra Research #FOS: Mathematics #Finite Group Theory Research #Geometric Topology (math.GT) #Geometric and Algebraic Topology #Primary 57M60 #Secondary 57R40

paper · pdf · doi:10.48550/arxiv.2208.14633

openalex publication_date 2022/08/31 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Suppose a finite group G acts on a manifold M. By a theorem of Mostow, also Palais, there is a G-equivariant embedding of M into the m-dimensional Euclidean space \RR^m for some m. We are interested in some explicit bounds of such m. First we provide an upper bound: there exists a G-equivariant embedding of M into \RR^d|G|+1, where |G| is the order of G and M embeds into \RR^d. Next we provide a lower bound for finite cyclic group action G: If there are l points having pairwise co-prime lengths of G-orbits greater than 1 and there is a G-equivariant embedding of M into \RR^m, then m≥ 2l. Some applications to surfaces are given.

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• Equivariant embeddings of Riemann surfaces in Euclidean spaces with minimal dimensions
• On equivariant isometric embeddings of Riemannian manifolds with symmetries

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