Totally umbilical hypersurfaces of product spaces
Given a Riemannian manifold M , and an open interval I ⊂ R , we characterize nontrivial totally umbilical hypersurfaces of the product M × I —as well as of warped products I × ω M —as those which are local graphs built on isoparametric families of totally umbilical hypersurfaces of M . By means of...
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| Veröffentlicht in: | Manuscripta mathematica 2022-11, Vol.169 (3-4), p.649-666 |
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| container_title | Manuscripta mathematica |
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| creator | de Lima, Ronaldo F. dos Santos, João Paulo |
| description | Given a Riemannian manifold
M
, and an open interval
I
⊂
R
,
we characterize nontrivial totally umbilical hypersurfaces of the product
M
×
I
—as well as of warped products
I
×
ω
M
—as those which are local graphs built on isoparametric families of totally umbilical hypersurfaces of
M
. By means of this characterization, we fully extend to
S
n
×
R
and
H
n
×
R
the results by Souam and Toubiana on the classification of totally umbilical surfaces in
S
2
×
R
and
H
2
×
R
.
It is also shown that an analogous classification holds for arbitrary warped products
I
×
ω
S
n
and
I
×
ω
H
n
. |
| doi_str_mv | 10.1007/s00229-021-01339-x |
| format | Article |
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M
, and an open interval
I
⊂
R
,
we characterize nontrivial totally umbilical hypersurfaces of the product
M
×
I
—as well as of warped products
I
×
ω
M
—as those which are local graphs built on isoparametric families of totally umbilical hypersurfaces of
M
. By means of this characterization, we fully extend to
S
n
×
R
and
H
n
×
R
the results by Souam and Toubiana on the classification of totally umbilical surfaces in
S
2
×
R
and
H
2
×
R
.
It is also shown that an analogous classification holds for arbitrary warped products
I
×
ω
S
n
and
I
×
ω
H
n
.</description><identifier>ISSN: 0025-2611</identifier><identifier>EISSN: 1432-1785</identifier><identifier>DOI: 10.1007/s00229-021-01339-x</identifier><language>eng</language><publisher>Berlin/Heidelberg: Springer Berlin Heidelberg</publisher><subject>Algebraic Geometry ; Calculus of Variations and Optimal Control; Optimization ; Classification ; Geometry ; Hyperspaces ; Lie Groups ; Mathematics ; Mathematics and Statistics ; Number Theory ; Riemann manifold ; Topological Groups</subject><ispartof>Manuscripta mathematica, 2022-11, Vol.169 (3-4), p.649-666</ispartof><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021</rights><rights>The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021.</rights><lds50>peer_reviewed</lds50><woscitedreferencessubscribed>false</woscitedreferencessubscribed><cites>FETCH-LOGICAL-c270t-77e404ff8db340518162b295acbf84a75e7c174469545cd515c2ad66011cf5483</cites><orcidid>0000-0002-7331-2721</orcidid></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Tsyndetics_thumb_exl</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/s00229-021-01339-x$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/s00229-021-01339-x$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>314,780,784,27924,27925,41488,42557,51319</link.rule.ids></links><search><creatorcontrib>de Lima, Ronaldo F.</creatorcontrib><creatorcontrib>dos Santos, João Paulo</creatorcontrib><title>Totally umbilical hypersurfaces of product spaces</title><title>Manuscripta mathematica</title><addtitle>manuscripta math</addtitle><description>Given a Riemannian manifold
M
, and an open interval
I
⊂
R
,
we characterize nontrivial totally umbilical hypersurfaces of the product
M
×
I
—as well as of warped products
I
×
ω
M
—as those which are local graphs built on isoparametric families of totally umbilical hypersurfaces of
M
. By means of this characterization, we fully extend to
S
n
×
R
and
H
n
×
R
the results by Souam and Toubiana on the classification of totally umbilical surfaces in
S
2
×
R
and
H
2
×
R
.
It is also shown that an analogous classification holds for arbitrary warped products
I
×
ω
S
n
and
I
×
ω
H
n
.</description><subject>Algebraic Geometry</subject><subject>Calculus of Variations and Optimal Control; Optimization</subject><subject>Classification</subject><subject>Geometry</subject><subject>Hyperspaces</subject><subject>Lie Groups</subject><subject>Mathematics</subject><subject>Mathematics and Statistics</subject><subject>Number Theory</subject><subject>Riemann manifold</subject><subject>Topological Groups</subject><issn>0025-2611</issn><issn>1432-1785</issn><fulltext>true</fulltext><rsrctype>article</rsrctype><creationdate>2022</creationdate><recordtype>article</recordtype><recordid>eNp9kE1LwzAYx4MoOKdfwFPBc_R58tK0RxnqhIGXeQ5pmmhHt9akhfXbm1nBm6cHnv8b_Ai5RbhHAPUQARgrKTCkgJyX9HhGFig4o6gKeU4WSZeU5YiX5CrGHUASFV8Q3HaDadspG_dV0zbWtNnn1LsQx-CNdTHrfNaHrh7tkMX-9LkmF9600d383iV5f37artZ08_byunrcUMsUDFQpJ0B4X9QVFyCxwJxVrJTGVr4QRkmnLCoh8lIKaWuJ0jJT5zkgWi9FwZfkbu5N81-ji4PedWM4pEnNFOMpwKVKLja7bOhiDM7rPjR7EyaNoE9o9IxGJzT6B40-phCfQzGZDx8u_FX_k_oGuA1lxA</recordid><startdate>20221101</startdate><enddate>20221101</enddate><creator>de Lima, Ronaldo F.</creator><creator>dos Santos, João Paulo</creator><general>Springer Berlin Heidelberg</general><general>Springer Nature B.V</general><scope>AAYXX</scope><scope>CITATION</scope><orcidid>https://orcid.org/0000-0002-7331-2721</orcidid></search><sort><creationdate>20221101</creationdate><title>Totally umbilical hypersurfaces of product spaces</title><author>de Lima, Ronaldo F. ; dos Santos, João Paulo</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-c270t-77e404ff8db340518162b295acbf84a75e7c174469545cd515c2ad66011cf5483</frbrgroupid><rsrctype>articles</rsrctype><prefilter>articles</prefilter><language>eng</language><creationdate>2022</creationdate><topic>Algebraic Geometry</topic><topic>Calculus of Variations and Optimal Control; Optimization</topic><topic>Classification</topic><topic>Geometry</topic><topic>Hyperspaces</topic><topic>Lie Groups</topic><topic>Mathematics</topic><topic>Mathematics and Statistics</topic><topic>Number Theory</topic><topic>Riemann manifold</topic><topic>Topological Groups</topic><toplevel>peer_reviewed</toplevel><toplevel>online_resources</toplevel><creatorcontrib>de Lima, Ronaldo F.</creatorcontrib><creatorcontrib>dos Santos, João Paulo</creatorcontrib><collection>CrossRef</collection><jtitle>Manuscripta mathematica</jtitle></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>de Lima, Ronaldo F.</au><au>dos Santos, João Paulo</au><format>journal</format><genre>article</genre><ristype>JOUR</ristype><atitle>Totally umbilical hypersurfaces of product spaces</atitle><jtitle>Manuscripta mathematica</jtitle><stitle>manuscripta math</stitle><date>2022-11-01</date><risdate>2022</risdate><volume>169</volume><issue>3-4</issue><spage>649</spage><epage>666</epage><pages>649-666</pages><issn>0025-2611</issn><eissn>1432-1785</eissn><abstract>Given a Riemannian manifold
M
, and an open interval
I
⊂
R
,
we characterize nontrivial totally umbilical hypersurfaces of the product
M
×
I
—as well as of warped products
I
×
ω
M
—as those which are local graphs built on isoparametric families of totally umbilical hypersurfaces of
M
. By means of this characterization, we fully extend to
S
n
×
R
and
H
n
×
R
the results by Souam and Toubiana on the classification of totally umbilical surfaces in
S
2
×
R
and
H
2
×
R
.
It is also shown that an analogous classification holds for arbitrary warped products
I
×
ω
S
n
and
I
×
ω
H
n
.</abstract><cop>Berlin/Heidelberg</cop><pub>Springer Berlin Heidelberg</pub><doi>10.1007/s00229-021-01339-x</doi><tpages>18</tpages><orcidid>https://orcid.org/0000-0002-7331-2721</orcidid></addata></record> |
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| language | eng |
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| source | SpringerLink Journals - AutoHoldings |
| subjects | Algebraic Geometry Calculus of Variations and Optimal Control Optimization Classification Geometry Hyperspaces Lie Groups Mathematics Mathematics and Statistics Number Theory Riemann manifold Topological Groups |
| title | Totally umbilical hypersurfaces of product spaces |
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