Hausdorff measures and dimensions in non equiregular sub-Riemannian manifolds
This paper is a starting point towards computing the Hausdorff dimension of submanifolds and the Hausdorff volume of small balls in a sub-Riemannian manifold with singular points. We first consider the case of a strongly equiregular submanifold, i. e., a smooth submanifold N for which the growth vec...
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| description | This paper is a starting point towards computing the Hausdorff dimension of submanifolds and the Hausdorff volume of small balls in a sub-Riemannian manifold with singular points. We first consider the case of a strongly equiregular submanifold, i. e., a smooth submanifold N for which the growth vector of the distribution D and the growth vector of the intersection of D with TN are constant on N. In this case, we generalize the result in [12], which relates the Hausdorff dimension to the growth vector of the distribution. We then consider analytic sub-Riemannian manifolds and, under the assumption that the singular point p is typical, we state a theorem which characterizes the Hausdorff dimension of the manifold and the finiteness of the Hausdorff volume of small balls B(p, ρ) in terms of the growth vector of both the distribution and the intersection of the distribution with the singular locus, and of the nonholonomic order at p of the volume form on M evaluated along some families of vector fields. |
| doi_str_mv | 10.1007/978-3-319-02132-4_13 |
| format | Book Chapter |
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We first consider the case of a strongly equiregular submanifold, i. e., a smooth submanifold N for which the growth vector of the distribution D and the growth vector of the intersection of D with TN are constant on N. In this case, we generalize the result in [12], which relates the Hausdorff dimension to the growth vector of the distribution. We then consider analytic sub-Riemannian manifolds and, under the assumption that the singular point p is typical, we state a theorem which characterizes the Hausdorff dimension of the manifold and the finiteness of the Hausdorff volume of small balls B(p, ρ) in terms of the growth vector of both the distribution and the intersection of the distribution with the singular locus, and of the nonholonomic order at p of the volume form on M evaluated along some families of vector fields.</description><identifier>ISSN: 2281-518X</identifier><identifier>ISBN: 3319021311</identifier><identifier>ISBN: 9783319021317</identifier><identifier>ISBN: 331902132X</identifier><identifier>ISBN: 9783319021324</identifier><identifier>EISSN: 2281-5198</identifier><identifier>EISBN: 331902132X</identifier><identifier>EISBN: 9783319021324</identifier><identifier>DOI: 10.1007/978-3-319-02132-4_13</identifier><identifier>OCLC: 892239514</identifier><identifier>LCCallNum: QA402.5-402.6</identifier><language>eng</language><publisher>Switzerland: Springer International Publishing AG</publisher><subject>Calculus of variations ; Differential & Riemannian geometry ; Differential Geometry ; Hausdorff Dimension ; Hausdorff Measure ; Infinite series ; Mathematics ; Metric Geometry ; Optimization and Control ; Regular Point ; Singular Point ; Tangent Cone</subject><ispartof>Geometric Control Theory and Sub-Riemannian Geometry, 2014, Vol.5, p.201-218</ispartof><rights>Springer International Publishing Switzerland 2014</rights><rights>Distributed under a Creative Commons Attribution 4.0 International License</rights><oa>free_for_read</oa><woscitedreferencessubscribed>false</woscitedreferencessubscribed><orcidid>0000-0001-8170-076X</orcidid><relation>Springer INdAM Series</relation></display><links><openurl>$$Topenurl_article</openurl><openurlfulltext>$$Topenurlfull_article</openurlfulltext><thumbnail>$$Uhttps://ebookcentral.proquest.com/covers/1782103-l.jpg</thumbnail><linktopdf>$$Uhttps://link.springer.com/content/pdf/10.1007/978-3-319-02132-4_13$$EPDF$$P50$$Gspringer$$H</linktopdf><linktohtml>$$Uhttps://link.springer.com/10.1007/978-3-319-02132-4_13$$EHTML$$P50$$Gspringer$$H</linktohtml><link.rule.ids>230,780,781,785,794,886,27930,38260,41447,42516</link.rule.ids><backlink>$$Uhttps://hal.science/hal-01137582$$DView record in HAL$$Hfree_for_read</backlink></links><search><contributor>Gauthier, Jean-Paul</contributor><contributor>Sarychev, Andrey</contributor><contributor>Stefani, Gianna</contributor><contributor>Sigalotti, Mario</contributor><contributor>Boscain, Ugo</contributor><contributor>Sarychev, Andrey</contributor><contributor>Boscain, Ugo</contributor><contributor>Sigalotti, Mario</contributor><contributor>Gauthier, Jean-Paul</contributor><contributor>Stefani, Gianna</contributor><creatorcontrib>Ghezzi, Roberta</creatorcontrib><creatorcontrib>Jean, Frédéric</creatorcontrib><title>Hausdorff measures and dimensions in non equiregular sub-Riemannian manifolds</title><title>Geometric Control Theory and Sub-Riemannian Geometry</title><description>This paper is a starting point towards computing the Hausdorff dimension of submanifolds and the Hausdorff volume of small balls in a sub-Riemannian manifold with singular points. 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We then consider analytic sub-Riemannian manifolds and, under the assumption that the singular point p is typical, we state a theorem which characterizes the Hausdorff dimension of the manifold and the finiteness of the Hausdorff volume of small balls B(p, ρ) in terms of the growth vector of both the distribution and the intersection of the distribution with the singular locus, and of the nonholonomic order at p of the volume form on M evaluated along some families of vector fields.</description><subject>Calculus of variations</subject><subject>Differential & Riemannian geometry</subject><subject>Differential Geometry</subject><subject>Hausdorff Dimension</subject><subject>Hausdorff Measure</subject><subject>Infinite series</subject><subject>Mathematics</subject><subject>Metric Geometry</subject><subject>Optimization and Control</subject><subject>Regular Point</subject><subject>Singular Point</subject><subject>Tangent Cone</subject><issn>2281-518X</issn><issn>2281-5198</issn><isbn>3319021311</isbn><isbn>9783319021317</isbn><isbn>331902132X</isbn><isbn>9783319021324</isbn><isbn>331902132X</isbn><isbn>9783319021324</isbn><fulltext>true</fulltext><rsrctype>book_chapter</rsrctype><creationdate>2014</creationdate><recordtype>book_chapter</recordtype><recordid>eNpFkEFP3DAQhU1pK7bAP-gh1x5MZ-wkto8ItSzSIiQEEjdrktis26yz2Buk_vt62aqcRvPmvdHMx9hXhAsEUN-N0lxyiYaDQCl4bVEesS-yKG_C0we2EEIjb9Do4_cB4sf_A_30mS20EUKaBusTdp7zLwBAoRpdNwt2u6Q5D1Pyvto4ynNyuaI4VEPYuJjDFHMVYhWnWLmXOST3PI-Uqjx3_D64DcUYKFalBj-NQz5jnzyN2Z3_q6fs8eePh6slX91d31xdrvhaCrnjxmhFpHpNvfGulW3tqYauaz202ntQRvle9uVI35lalr4B8DRQ04IQnZKn7Nth75pGu01hQ-mPnSjY5eXK7jVAlOVD8YrFKw7eXIzx2SXbTdPvbBHsnrItlK20BZ19g2r3lEuoPoS2aXqZXd5Zt0_1Lu4Sjf2atjuXyg6lBYK02GgroJF_AWBce0M</recordid><startdate>20140101</startdate><enddate>20140101</enddate><creator>Ghezzi, Roberta</creator><creator>Jean, Frédéric</creator><general>Springer International Publishing AG</general><general>Springer International Publishing</general><scope>FFUUA</scope><scope>1XC</scope><orcidid>https://orcid.org/0000-0001-8170-076X</orcidid></search><sort><creationdate>20140101</creationdate><title>Hausdorff measures and dimensions in non equiregular sub-Riemannian manifolds</title><author>Ghezzi, Roberta ; Jean, Frédéric</author></sort><facets><frbrtype>5</frbrtype><frbrgroupid>cdi_FETCH-LOGICAL-h323t-9987aa7c8ac9fe6364fa40bb6f068ff0797fc3c127fb943079500fada56022b73</frbrgroupid><rsrctype>book_chapters</rsrctype><prefilter>book_chapters</prefilter><language>eng</language><creationdate>2014</creationdate><topic>Calculus of variations</topic><topic>Differential & Riemannian geometry</topic><topic>Differential Geometry</topic><topic>Hausdorff Dimension</topic><topic>Hausdorff Measure</topic><topic>Infinite series</topic><topic>Mathematics</topic><topic>Metric Geometry</topic><topic>Optimization and Control</topic><topic>Regular Point</topic><topic>Singular Point</topic><topic>Tangent Cone</topic><toplevel>online_resources</toplevel><creatorcontrib>Ghezzi, Roberta</creatorcontrib><creatorcontrib>Jean, Frédéric</creatorcontrib><collection>ProQuest Ebook Central - Book Chapters - Demo use only</collection><collection>Hyper Article en Ligne (HAL)</collection></facets><delivery><delcategory>Remote Search Resource</delcategory><fulltext>fulltext</fulltext></delivery><addata><au>Ghezzi, Roberta</au><au>Jean, Frédéric</au><au>Gauthier, Jean-Paul</au><au>Sarychev, Andrey</au><au>Stefani, Gianna</au><au>Sigalotti, Mario</au><au>Boscain, Ugo</au><au>Sarychev, Andrey</au><au>Boscain, Ugo</au><au>Sigalotti, Mario</au><au>Gauthier, Jean-Paul</au><au>Stefani, Gianna</au><format>book</format><genre>bookitem</genre><ristype>CHAP</ristype><atitle>Hausdorff measures and dimensions in non equiregular sub-Riemannian manifolds</atitle><btitle>Geometric Control Theory and Sub-Riemannian Geometry</btitle><seriestitle>Springer INdAM Series</seriestitle><date>2014-01-01</date><risdate>2014</risdate><volume>5</volume><spage>201</spage><epage>218</epage><pages>201-218</pages><issn>2281-518X</issn><eissn>2281-5198</eissn><isbn>3319021311</isbn><isbn>9783319021317</isbn><isbn>331902132X</isbn><isbn>9783319021324</isbn><eisbn>331902132X</eisbn><eisbn>9783319021324</eisbn><abstract>This paper is a starting point towards computing the Hausdorff dimension of submanifolds and the Hausdorff volume of small balls in a sub-Riemannian manifold with singular points. We first consider the case of a strongly equiregular submanifold, i. e., a smooth submanifold N for which the growth vector of the distribution D and the growth vector of the intersection of D with TN are constant on N. In this case, we generalize the result in [12], which relates the Hausdorff dimension to the growth vector of the distribution. We then consider analytic sub-Riemannian manifolds and, under the assumption that the singular point p is typical, we state a theorem which characterizes the Hausdorff dimension of the manifold and the finiteness of the Hausdorff volume of small balls B(p, ρ) in terms of the growth vector of both the distribution and the intersection of the distribution with the singular locus, and of the nonholonomic order at p of the volume form on M evaluated along some families of vector fields.</abstract><cop>Switzerland</cop><pub>Springer International Publishing AG</pub><doi>10.1007/978-3-319-02132-4_13</doi><oclcid>892239514</oclcid><tpages>18</tpages><orcidid>https://orcid.org/0000-0001-8170-076X</orcidid><oa>free_for_read</oa></addata></record> |
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| ispartof | Geometric Control Theory and Sub-Riemannian Geometry, 2014, Vol.5, p.201-218 |
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| language | eng |
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| source | Springer Books |
| subjects | Calculus of variations Differential & Riemannian geometry Differential Geometry Hausdorff Dimension Hausdorff Measure Infinite series Mathematics Metric Geometry Optimization and Control Regular Point Singular Point Tangent Cone |
| title | Hausdorff measures and dimensions in non equiregular sub-Riemannian manifolds |
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