Some properties of abnormal extremals on Lie groups
Abstract Let G be a connected Lie group and D be a bracket generating left invariant distribution. In this paper, first, we prove that all sub-Riemannian minimizers are smooth in Lie groups if the distribution D satisfies [D, [D,D]] ⊂ D and [K, [K,K]] ⊂ K for any proper sub-distribution K of D. Seco...
Ausführliche Beschreibung
Autor*in: |
Huang, Ti Ren [verfasserIn] |
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Sprache: |
Englisch |
Erschienen: |
2014 |
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Anmerkung: |
© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2014 |
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Übergeordnetes Werk: |
Enthalten in: Acta mathematica sinica - Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society, 1985, 30(2014), 12 vom: 15. Nov., Seite 2119-2136 |
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Übergeordnetes Werk: |
volume:30 ; year:2014 ; number:12 ; day:15 ; month:11 ; pages:2119-2136 |
Links: |
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DOI / URN: |
10.1007/s10114-014-1286-9 |
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Katalog-ID: |
OLC2062519370 |
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10.1007/s10114-014-1286-9 doi (DE-627)OLC2062519370 (DE-He213)s10114-014-1286-9-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Huang, Ti Ren verfasserin aut Some properties of abnormal extremals on Lie groups 2014 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2014 Abstract Let G be a connected Lie group and D be a bracket generating left invariant distribution. In this paper, first, we prove that all sub-Riemannian minimizers are smooth in Lie groups if the distribution D satisfies [D, [D,D]] ⊂ D and [K, [K,K]] ⊂ K for any proper sub-distribution K of D. Second, we prove that all sub-Riemannian minimizers are smooth in Lie group if the rank-3 distribution D satisfies Condition (B). Third, we discuss characterizations of normal extremals, abnormal extremals, rigid curves, and minimizers on product sub-riemannian Lie groups. We prove that not all strictly abnormal minimizers are rigid curves and construct a strictly abnormal minimizer which is not a rigid curve. Extremal geodesic rigid curve Lie group Yang, Xiao Ping aut Enthalten in Acta mathematica sinica Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society, 1985 30(2014), 12 vom: 15. Nov., Seite 2119-2136 (DE-627)129236772 (DE-600)58083-1 (DE-576)091206189 1000-9574 nnns volume:30 year:2014 number:12 day:15 month:11 pages:2119-2136 https://doi.org/10.1007/s10114-014-1286-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_2018 GBV_ILN_2088 AR 30 2014 12 15 11 2119-2136 |
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10.1007/s10114-014-1286-9 doi (DE-627)OLC2062519370 (DE-He213)s10114-014-1286-9-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Huang, Ti Ren verfasserin aut Some properties of abnormal extremals on Lie groups 2014 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2014 Abstract Let G be a connected Lie group and D be a bracket generating left invariant distribution. In this paper, first, we prove that all sub-Riemannian minimizers are smooth in Lie groups if the distribution D satisfies [D, [D,D]] ⊂ D and [K, [K,K]] ⊂ K for any proper sub-distribution K of D. Second, we prove that all sub-Riemannian minimizers are smooth in Lie group if the rank-3 distribution D satisfies Condition (B). Third, we discuss characterizations of normal extremals, abnormal extremals, rigid curves, and minimizers on product sub-riemannian Lie groups. We prove that not all strictly abnormal minimizers are rigid curves and construct a strictly abnormal minimizer which is not a rigid curve. Extremal geodesic rigid curve Lie group Yang, Xiao Ping aut Enthalten in Acta mathematica sinica Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society, 1985 30(2014), 12 vom: 15. Nov., Seite 2119-2136 (DE-627)129236772 (DE-600)58083-1 (DE-576)091206189 1000-9574 nnns volume:30 year:2014 number:12 day:15 month:11 pages:2119-2136 https://doi.org/10.1007/s10114-014-1286-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_2018 GBV_ILN_2088 AR 30 2014 12 15 11 2119-2136 |
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10.1007/s10114-014-1286-9 doi (DE-627)OLC2062519370 (DE-He213)s10114-014-1286-9-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Huang, Ti Ren verfasserin aut Some properties of abnormal extremals on Lie groups 2014 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2014 Abstract Let G be a connected Lie group and D be a bracket generating left invariant distribution. In this paper, first, we prove that all sub-Riemannian minimizers are smooth in Lie groups if the distribution D satisfies [D, [D,D]] ⊂ D and [K, [K,K]] ⊂ K for any proper sub-distribution K of D. Second, we prove that all sub-Riemannian minimizers are smooth in Lie group if the rank-3 distribution D satisfies Condition (B). Third, we discuss characterizations of normal extremals, abnormal extremals, rigid curves, and minimizers on product sub-riemannian Lie groups. We prove that not all strictly abnormal minimizers are rigid curves and construct a strictly abnormal minimizer which is not a rigid curve. Extremal geodesic rigid curve Lie group Yang, Xiao Ping aut Enthalten in Acta mathematica sinica Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society, 1985 30(2014), 12 vom: 15. Nov., Seite 2119-2136 (DE-627)129236772 (DE-600)58083-1 (DE-576)091206189 1000-9574 nnns volume:30 year:2014 number:12 day:15 month:11 pages:2119-2136 https://doi.org/10.1007/s10114-014-1286-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_2018 GBV_ILN_2088 AR 30 2014 12 15 11 2119-2136 |
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10.1007/s10114-014-1286-9 doi (DE-627)OLC2062519370 (DE-He213)s10114-014-1286-9-p DE-627 ger DE-627 rakwb eng 510 VZ 510 VZ 17,1 ssgn Huang, Ti Ren verfasserin aut Some properties of abnormal extremals on Lie groups 2014 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2014 Abstract Let G be a connected Lie group and D be a bracket generating left invariant distribution. In this paper, first, we prove that all sub-Riemannian minimizers are smooth in Lie groups if the distribution D satisfies [D, [D,D]] ⊂ D and [K, [K,K]] ⊂ K for any proper sub-distribution K of D. Second, we prove that all sub-Riemannian minimizers are smooth in Lie group if the rank-3 distribution D satisfies Condition (B). Third, we discuss characterizations of normal extremals, abnormal extremals, rigid curves, and minimizers on product sub-riemannian Lie groups. We prove that not all strictly abnormal minimizers are rigid curves and construct a strictly abnormal minimizer which is not a rigid curve. Extremal geodesic rigid curve Lie group Yang, Xiao Ping aut Enthalten in Acta mathematica sinica Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society, 1985 30(2014), 12 vom: 15. Nov., Seite 2119-2136 (DE-627)129236772 (DE-600)58083-1 (DE-576)091206189 1000-9574 nnns volume:30 year:2014 number:12 day:15 month:11 pages:2119-2136 https://doi.org/10.1007/s10114-014-1286-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_40 GBV_ILN_70 GBV_ILN_2018 GBV_ILN_2088 AR 30 2014 12 15 11 2119-2136 |
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Abstract Let G be a connected Lie group and D be a bracket generating left invariant distribution. In this paper, first, we prove that all sub-Riemannian minimizers are smooth in Lie groups if the distribution D satisfies [D, [D,D]] ⊂ D and [K, [K,K]] ⊂ K for any proper sub-distribution K of D. Second, we prove that all sub-Riemannian minimizers are smooth in Lie group if the rank-3 distribution D satisfies Condition (B). Third, we discuss characterizations of normal extremals, abnormal extremals, rigid curves, and minimizers on product sub-riemannian Lie groups. We prove that not all strictly abnormal minimizers are rigid curves and construct a strictly abnormal minimizer which is not a rigid curve. © Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2014 |
abstractGer |
Abstract Let G be a connected Lie group and D be a bracket generating left invariant distribution. In this paper, first, we prove that all sub-Riemannian minimizers are smooth in Lie groups if the distribution D satisfies [D, [D,D]] ⊂ D and [K, [K,K]] ⊂ K for any proper sub-distribution K of D. Second, we prove that all sub-Riemannian minimizers are smooth in Lie group if the rank-3 distribution D satisfies Condition (B). Third, we discuss characterizations of normal extremals, abnormal extremals, rigid curves, and minimizers on product sub-riemannian Lie groups. We prove that not all strictly abnormal minimizers are rigid curves and construct a strictly abnormal minimizer which is not a rigid curve. © Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2014 |
abstract_unstemmed |
Abstract Let G be a connected Lie group and D be a bracket generating left invariant distribution. In this paper, first, we prove that all sub-Riemannian minimizers are smooth in Lie groups if the distribution D satisfies [D, [D,D]] ⊂ D and [K, [K,K]] ⊂ K for any proper sub-distribution K of D. Second, we prove that all sub-Riemannian minimizers are smooth in Lie group if the rank-3 distribution D satisfies Condition (B). Third, we discuss characterizations of normal extremals, abnormal extremals, rigid curves, and minimizers on product sub-riemannian Lie groups. We prove that not all strictly abnormal minimizers are rigid curves and construct a strictly abnormal minimizer which is not a rigid curve. © Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2014 |
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<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">OLC2062519370</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20230502172617.0</controlfield><controlfield tag="007">tu</controlfield><controlfield tag="008">200820s2014 xx ||||| 00| ||eng c</controlfield><datafield tag="024" ind1="7" ind2=" "><subfield code="a">10.1007/s10114-014-1286-9</subfield><subfield code="2">doi</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-627)OLC2062519370</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(DE-He213)s10114-014-1286-9-p</subfield></datafield><datafield tag="040" ind1=" " ind2=" "><subfield code="a">DE-627</subfield><subfield code="b">ger</subfield><subfield code="c">DE-627</subfield><subfield code="e">rakwb</subfield></datafield><datafield tag="041" ind1=" " ind2=" "><subfield code="a">eng</subfield></datafield><datafield tag="082" ind1="0" ind2="4"><subfield code="a">510</subfield><subfield code="q">VZ</subfield></datafield><datafield tag="082" ind1="0" ind2="4"><subfield code="a">510</subfield><subfield code="q">VZ</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">17,1</subfield><subfield code="2">ssgn</subfield></datafield><datafield tag="100" ind1="1" ind2=" "><subfield code="a">Huang, Ti Ren</subfield><subfield code="e">verfasserin</subfield><subfield code="4">aut</subfield></datafield><datafield tag="245" ind1="1" ind2="0"><subfield code="a">Some properties of abnormal extremals on Lie groups</subfield></datafield><datafield tag="264" ind1=" " ind2="1"><subfield code="c">2014</subfield></datafield><datafield tag="336" ind1=" " ind2=" "><subfield code="a">Text</subfield><subfield code="b">txt</subfield><subfield code="2">rdacontent</subfield></datafield><datafield tag="337" ind1=" " ind2=" "><subfield code="a">ohne Hilfsmittel zu benutzen</subfield><subfield code="b">n</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Band</subfield><subfield code="b">nc</subfield><subfield code="2">rdacarrier</subfield></datafield><datafield tag="500" ind1=" " ind2=" "><subfield code="a">© Institute of Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Chinese Mathematical Society and Springer-Verlag Berlin Heidelberg 2014</subfield></datafield><datafield tag="520" ind1=" " ind2=" "><subfield code="a">Abstract Let G be a connected Lie group and D be a bracket generating left invariant distribution. In this paper, first, we prove that all sub-Riemannian minimizers are smooth in Lie groups if the distribution D satisfies [D, [D,D]] ⊂ D and [K, [K,K]] ⊂ K for any proper sub-distribution K of D. Second, we prove that all sub-Riemannian minimizers are smooth in Lie group if the rank-3 distribution D satisfies Condition (B). Third, we discuss characterizations of normal extremals, abnormal extremals, rigid curves, and minimizers on product sub-riemannian Lie groups. We prove that not all strictly abnormal minimizers are rigid curves and construct a strictly abnormal minimizer which is not a rigid curve.</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Extremal</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">geodesic</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">rigid curve</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Lie group</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Yang, Xiao Ping</subfield><subfield code="4">aut</subfield></datafield><datafield tag="773" ind1="0" ind2="8"><subfield code="i">Enthalten in</subfield><subfield code="t">Acta mathematica sinica</subfield><subfield code="d">Institute of Mathematics, Chinese Academy of Sciences and Chinese Mathematical Society, 1985</subfield><subfield code="g">30(2014), 12 vom: 15. 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