On the geometry of Riemannian isometric embeddings
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866918202825506816 |
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| author | Burago, Dmitri Qiu, Hongda |
| author_facet | Burago, Dmitri Qiu, Hongda |
| contents | This note pertains to isometric embeddings endowed with certain geometric properties. We study two embedding problems for a Riemannian manifold $M$ which is diffeomorphic to $\RR^n$ and admits a Bieberbach group $Γ$ acting by isometries. The first problem concerns the existence of an isometric embedding of $M$ into a bounded subset of some Euclidean space $\RR^{D_1}$. The second problem seeks a $Γ$-equivariant isometric embdding of $M$ into $\RR^{D_2}$. By using a known trick in a novel way, our idea yields results with $D_1 = N+2n$ and $D_2 = N+n$, where $N$ is the Nash dimension of $ M/Γ$. Moreover, we also show that an $n$-dimensional smooth manifold, of Nash dimension $N$, can be isometrically embedded into a bounded subset of $\RR^{2N}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2507_23164 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the geometry of Riemannian isometric embeddings Burago, Dmitri Qiu, Hongda Differential Geometry This note pertains to isometric embeddings endowed with certain geometric properties. We study two embedding problems for a Riemannian manifold $M$ which is diffeomorphic to $\RR^n$ and admits a Bieberbach group $Γ$ acting by isometries. The first problem concerns the existence of an isometric embedding of $M$ into a bounded subset of some Euclidean space $\RR^{D_1}$. The second problem seeks a $Γ$-equivariant isometric embdding of $M$ into $\RR^{D_2}$. By using a known trick in a novel way, our idea yields results with $D_1 = N+2n$ and $D_2 = N+n$, where $N$ is the Nash dimension of $ M/Γ$. Moreover, we also show that an $n$-dimensional smooth manifold, of Nash dimension $N$, can be isometrically embedded into a bounded subset of $\RR^{2N}$. |
| title | On the geometry of Riemannian isometric embeddings |
| topic | Differential Geometry |
| url | https://arxiv.org/abs/2507.23164 |