October 9, 2025Open Access

A remark on equivariant Riemannian isometric embeddings preserving symmetries

Key Points

The study reveals new isometric embedding results for Riemannian tori, enhancing understanding of their geometric properties.
Results indicate that embeddings can be achieved in Euclidean spaces with dimensions given by D1 = N + 2n and D2 = N + n.
The method utilized provides a novel application of existing techniques in equivariant embeddings concerning covering maps.
Doubts remain about the optimality of the derived dimensional bounds, suggesting further inquiry is needed.

Abstract

This remark pertains to isometric embeddings endowed with certain geometric properties. We study two embeddings problems for the universal cover M of an n-dimensional Riemannian torus (ⁿ, g). The first concerns the existence of an isometric embedding of M into a bounded subset of some Euclidean space ^D₁, and the second one seeks an isometric embdding of M that is equivariant with respect to the deck transformation group of covering map. By using a known trick in a novel way, our idea yields results with D₁ = N+2n and D₂ = N+n, where N is the Nash dimension of ⁿ. However, we doubt whether these bounds are optimal.

A remark on equivariant Riemannian isometric embeddings preserving symmetries

Key Points

Abstract

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