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

Gespeichert in:

Autor*in:	Huang, Ti Ren [verfasserIn] Yang, Xiao Ping [verfasserIn]

Format:	E-Artikel
Sprache:	Englisch

Erschienen:	2014

Schlagwörter:	Extremal geodesic rigid curve Lie group

Übergeordnetes Werk:	Enthalten in: Acta mathematica sinica - Berlin : Springer, 1985, 30(2014), 12 vom: 15. Nov., Seite 2119-2136
Übergeordnetes Werk:	volume:30 ; year:2014 ; number:12 ; day:15 ; month:11 ; pages:2119-2136

Links:	Volltext

DOI / URN:	10.1007/s10114-014-1286-9

Katalog-ID:	SPR009018034

Internformat


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520			\|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.
650		4	\|a Extremal \|7 (dpeaa)DE-He213
650		4	\|a geodesic \|7 (dpeaa)DE-He213
650		4	\|a rigid curve \|7 (dpeaa)DE-He213
650		4	\|a Lie group \|7 (dpeaa)DE-He213
700	1		\|a Yang, Xiao Ping \|e verfasserin \|4 aut
773	0	8	\|i Enthalten in \|t Acta mathematica sinica \|d Berlin : Springer, 1985 \|g 30(2014), 12 vom: 15. Nov., Seite 2119-2136 \|w (DE-627)312226470 \|w (DE-600)2011055-8 \|x 1439-7617 \|7 nnns
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856	4	0	\|u https://dx.doi.org/10.1007/s10114-014-1286-9 \|z lizenzpflichtig \|3 Volltext
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912			\|a GBV_ILN_20
912			\|a GBV_ILN_22
912			\|a GBV_ILN_23
912			\|a GBV_ILN_24
912			\|a GBV_ILN_31
912			\|a GBV_ILN_32
912			\|a GBV_ILN_39
912			\|a GBV_ILN_40
912			\|a GBV_ILN_60
912			\|a GBV_ILN_62
912			\|a GBV_ILN_63
912			\|a GBV_ILN_69
912			\|a GBV_ILN_70
912			\|a GBV_ILN_73
912			\|a GBV_ILN_74
912			\|a GBV_ILN_90
912			\|a GBV_ILN_95
912			\|a GBV_ILN_100
912			\|a GBV_ILN_105
912			\|a GBV_ILN_110
912			\|a GBV_ILN_120
912			\|a GBV_ILN_138
912			\|a GBV_ILN_150
912			\|a GBV_ILN_151
912			\|a GBV_ILN_152
912			\|a GBV_ILN_161
912			\|a GBV_ILN_170
912			\|a GBV_ILN_171
912			\|a GBV_ILN_187
912			\|a GBV_ILN_206
912			\|a GBV_ILN_213
912			\|a GBV_ILN_224
912			\|a GBV_ILN_230
912			\|a GBV_ILN_250
912			\|a GBV_ILN_281
912			\|a GBV_ILN_285
912			\|a GBV_ILN_293
912			\|a GBV_ILN_370
912			\|a GBV_ILN_602
912			\|a GBV_ILN_636
912			\|a GBV_ILN_702
912			\|a GBV_ILN_2001
912			\|a GBV_ILN_2003
912			\|a GBV_ILN_2004
912			\|a GBV_ILN_2005
912			\|a GBV_ILN_2006
912			\|a GBV_ILN_2007
912			\|a GBV_ILN_2008
912			\|a GBV_ILN_2009
912			\|a GBV_ILN_2010
912			\|a GBV_ILN_2011
912			\|a GBV_ILN_2014
912			\|a GBV_ILN_2015
912			\|a GBV_ILN_2020
912			\|a GBV_ILN_2021
912			\|a GBV_ILN_2025
912			\|a GBV_ILN_2026
912			\|a GBV_ILN_2027
912			\|a GBV_ILN_2031
912			\|a GBV_ILN_2034
912			\|a GBV_ILN_2037
912			\|a GBV_ILN_2038
912			\|a GBV_ILN_2039
912			\|a GBV_ILN_2044
912			\|a GBV_ILN_2048
912			\|a GBV_ILN_2049
912			\|a GBV_ILN_2050
912			\|a GBV_ILN_2055
912			\|a GBV_ILN_2056
912			\|a GBV_ILN_2057
912			\|a GBV_ILN_2059
912			\|a GBV_ILN_2061
912			\|a GBV_ILN_2064
912			\|a GBV_ILN_2065
912			\|a GBV_ILN_2068
912			\|a GBV_ILN_2070
912			\|a GBV_ILN_2086
912			\|a GBV_ILN_2088
912			\|a GBV_ILN_2093
912			\|a GBV_ILN_2106
912			\|a GBV_ILN_2107
912			\|a GBV_ILN_2108
912			\|a GBV_ILN_2110
912			\|a GBV_ILN_2111
912			\|a GBV_ILN_2112
912			\|a GBV_ILN_2113
912			\|a GBV_ILN_2118
912			\|a GBV_ILN_2119
912			\|a GBV_ILN_2122
912			\|a GBV_ILN_2129
912			\|a GBV_ILN_2143
912			\|a GBV_ILN_2144
912			\|a GBV_ILN_2147
912			\|a GBV_ILN_2148
912			\|a GBV_ILN_2152
912			\|a GBV_ILN_2153
912			\|a GBV_ILN_2188
912			\|a GBV_ILN_2190
912			\|a GBV_ILN_2232
912			\|a GBV_ILN_2336
912			\|a GBV_ILN_2446
912			\|a GBV_ILN_2470
912			\|a GBV_ILN_2472
912			\|a GBV_ILN_2507
912			\|a GBV_ILN_2522
912			\|a GBV_ILN_2548
912			\|a GBV_ILN_4035
912			\|a GBV_ILN_4037
912			\|a GBV_ILN_4046
912			\|a GBV_ILN_4112
912			\|a GBV_ILN_4125
912			\|a GBV_ILN_4126
912			\|a GBV_ILN_4242
912			\|a GBV_ILN_4246
912			\|a GBV_ILN_4249
912			\|a GBV_ILN_4251
912			\|a GBV_ILN_4305
912			\|a GBV_ILN_4306
912			\|a GBV_ILN_4307
912			\|a GBV_ILN_4313
912			\|a GBV_ILN_4322
912			\|a GBV_ILN_4323
912			\|a GBV_ILN_4324
912			\|a GBV_ILN_4325
912			\|a GBV_ILN_4326
912			\|a GBV_ILN_4333
912			\|a GBV_ILN_4334
912			\|a GBV_ILN_4335
912			\|a GBV_ILN_4336
912			\|a GBV_ILN_4338
912			\|a GBV_ILN_4393
912			\|a GBV_ILN_4700
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Indexfelder

author_variant	t r h tr trh x p y xp xpy
matchkey_str	article:14397617:2014----::oerprisfbomlxrml
hierarchy_sort_str	2014
bklnumber	31.00
publishDate	2014
allfields	10.1007/s10114-014-1286-9 doi (DE-627)SPR009018034 (SPR)s10114-014-1286-9-e DE-627 ger DE-627 rakwb eng 510 ASE 510 ASE 31.00 bkl Huang, Ti Ren verfasserin aut Some properties of abnormal extremals on Lie groups 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier 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 (dpeaa)DE-He213 geodesic (dpeaa)DE-He213 rigid curve (dpeaa)DE-He213 Lie group (dpeaa)DE-He213 Yang, Xiao Ping verfasserin aut Enthalten in Acta mathematica sinica Berlin : Springer, 1985 30(2014), 12 vom: 15. Nov., Seite 2119-2136 (DE-627)312226470 (DE-600)2011055-8 1439-7617 nnns volume:30 year:2014 number:12 day:15 month:11 pages:2119-2136 https://dx.doi.org/10.1007/s10114-014-1286-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.00 ASE AR 30 2014 12 15 11 2119-2136
spelling	10.1007/s10114-014-1286-9 doi (DE-627)SPR009018034 (SPR)s10114-014-1286-9-e DE-627 ger DE-627 rakwb eng 510 ASE 510 ASE 31.00 bkl Huang, Ti Ren verfasserin aut Some properties of abnormal extremals on Lie groups 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier 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 (dpeaa)DE-He213 geodesic (dpeaa)DE-He213 rigid curve (dpeaa)DE-He213 Lie group (dpeaa)DE-He213 Yang, Xiao Ping verfasserin aut Enthalten in Acta mathematica sinica Berlin : Springer, 1985 30(2014), 12 vom: 15. Nov., Seite 2119-2136 (DE-627)312226470 (DE-600)2011055-8 1439-7617 nnns volume:30 year:2014 number:12 day:15 month:11 pages:2119-2136 https://dx.doi.org/10.1007/s10114-014-1286-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.00 ASE AR 30 2014 12 15 11 2119-2136
allfields_unstemmed	10.1007/s10114-014-1286-9 doi (DE-627)SPR009018034 (SPR)s10114-014-1286-9-e DE-627 ger DE-627 rakwb eng 510 ASE 510 ASE 31.00 bkl Huang, Ti Ren verfasserin aut Some properties of abnormal extremals on Lie groups 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier 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 (dpeaa)DE-He213 geodesic (dpeaa)DE-He213 rigid curve (dpeaa)DE-He213 Lie group (dpeaa)DE-He213 Yang, Xiao Ping verfasserin aut Enthalten in Acta mathematica sinica Berlin : Springer, 1985 30(2014), 12 vom: 15. Nov., Seite 2119-2136 (DE-627)312226470 (DE-600)2011055-8 1439-7617 nnns volume:30 year:2014 number:12 day:15 month:11 pages:2119-2136 https://dx.doi.org/10.1007/s10114-014-1286-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.00 ASE AR 30 2014 12 15 11 2119-2136
allfieldsGer	10.1007/s10114-014-1286-9 doi (DE-627)SPR009018034 (SPR)s10114-014-1286-9-e DE-627 ger DE-627 rakwb eng 510 ASE 510 ASE 31.00 bkl Huang, Ti Ren verfasserin aut Some properties of abnormal extremals on Lie groups 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier 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 (dpeaa)DE-He213 geodesic (dpeaa)DE-He213 rigid curve (dpeaa)DE-He213 Lie group (dpeaa)DE-He213 Yang, Xiao Ping verfasserin aut Enthalten in Acta mathematica sinica Berlin : Springer, 1985 30(2014), 12 vom: 15. Nov., Seite 2119-2136 (DE-627)312226470 (DE-600)2011055-8 1439-7617 nnns volume:30 year:2014 number:12 day:15 month:11 pages:2119-2136 https://dx.doi.org/10.1007/s10114-014-1286-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.00 ASE AR 30 2014 12 15 11 2119-2136
allfieldsSound	10.1007/s10114-014-1286-9 doi (DE-627)SPR009018034 (SPR)s10114-014-1286-9-e DE-627 ger DE-627 rakwb eng 510 ASE 510 ASE 31.00 bkl Huang, Ti Ren verfasserin aut Some properties of abnormal extremals on Lie groups 2014 Text txt rdacontent Computermedien c rdamedia Online-Ressource cr rdacarrier 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 (dpeaa)DE-He213 geodesic (dpeaa)DE-He213 rigid curve (dpeaa)DE-He213 Lie group (dpeaa)DE-He213 Yang, Xiao Ping verfasserin aut Enthalten in Acta mathematica sinica Berlin : Springer, 1985 30(2014), 12 vom: 15. Nov., Seite 2119-2136 (DE-627)312226470 (DE-600)2011055-8 1439-7617 nnns volume:30 year:2014 number:12 day:15 month:11 pages:2119-2136 https://dx.doi.org/10.1007/s10114-014-1286-9 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_SPRINGER SSG-OPC-MAT SSG-OPC-ASE GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_23 GBV_ILN_24 GBV_ILN_31 GBV_ILN_32 GBV_ILN_39 GBV_ILN_40 GBV_ILN_60 GBV_ILN_62 GBV_ILN_63 GBV_ILN_69 GBV_ILN_70 GBV_ILN_73 GBV_ILN_74 GBV_ILN_90 GBV_ILN_95 GBV_ILN_100 GBV_ILN_105 GBV_ILN_110 GBV_ILN_120 GBV_ILN_138 GBV_ILN_150 GBV_ILN_151 GBV_ILN_152 GBV_ILN_161 GBV_ILN_170 GBV_ILN_171 GBV_ILN_187 GBV_ILN_206 GBV_ILN_213 GBV_ILN_224 GBV_ILN_230 GBV_ILN_250 GBV_ILN_281 GBV_ILN_285 GBV_ILN_293 GBV_ILN_370 GBV_ILN_602 GBV_ILN_636 GBV_ILN_702 GBV_ILN_2001 GBV_ILN_2003 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2006 GBV_ILN_2007 GBV_ILN_2008 GBV_ILN_2009 GBV_ILN_2010 GBV_ILN_2011 GBV_ILN_2014 GBV_ILN_2015 GBV_ILN_2020 GBV_ILN_2021 GBV_ILN_2025 GBV_ILN_2026 GBV_ILN_2027 GBV_ILN_2031 GBV_ILN_2034 GBV_ILN_2037 GBV_ILN_2038 GBV_ILN_2039 GBV_ILN_2044 GBV_ILN_2048 GBV_ILN_2049 GBV_ILN_2050 GBV_ILN_2055 GBV_ILN_2056 GBV_ILN_2057 GBV_ILN_2059 GBV_ILN_2061 GBV_ILN_2064 GBV_ILN_2065 GBV_ILN_2068 GBV_ILN_2070 GBV_ILN_2086 GBV_ILN_2088 GBV_ILN_2093 GBV_ILN_2106 GBV_ILN_2107 GBV_ILN_2108 GBV_ILN_2110 GBV_ILN_2111 GBV_ILN_2112 GBV_ILN_2113 GBV_ILN_2118 GBV_ILN_2119 GBV_ILN_2122 GBV_ILN_2129 GBV_ILN_2143 GBV_ILN_2144 GBV_ILN_2147 GBV_ILN_2148 GBV_ILN_2152 GBV_ILN_2153 GBV_ILN_2188 GBV_ILN_2190 GBV_ILN_2232 GBV_ILN_2336 GBV_ILN_2446 GBV_ILN_2470 GBV_ILN_2472 GBV_ILN_2507 GBV_ILN_2522 GBV_ILN_2548 GBV_ILN_4035 GBV_ILN_4037 GBV_ILN_4046 GBV_ILN_4112 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4242 GBV_ILN_4246 GBV_ILN_4249 GBV_ILN_4251 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4322 GBV_ILN_4323 GBV_ILN_4324 GBV_ILN_4325 GBV_ILN_4326 GBV_ILN_4333 GBV_ILN_4334 GBV_ILN_4335 GBV_ILN_4336 GBV_ILN_4338 GBV_ILN_4393 GBV_ILN_4700 31.00 ASE AR 30 2014 12 15 11 2119-2136
language	English
source	Enthalten in Acta mathematica sinica 30(2014), 12 vom: 15. Nov., Seite 2119-2136 volume:30 year:2014 number:12 day:15 month:11 pages:2119-2136
sourceStr	Enthalten in Acta mathematica sinica 30(2014), 12 vom: 15. Nov., Seite 2119-2136 volume:30 year:2014 number:12 day:15 month:11 pages:2119-2136
format_phy_str_mv	Article
institution	findex.gbv.de
topic_facet	Extremal geodesic rigid curve Lie group
dewey-raw	510
isfreeaccess_bool	false
container_title	Acta mathematica sinica
authorswithroles_txt_mv	Huang, Ti Ren @@aut@@ Yang, Xiao Ping @@aut@@
publishDateDaySort_date	2014-11-15T00:00:00Z
hierarchy_top_id	312226470
dewey-sort	3510
id	SPR009018034
language_de	englisch
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author	Huang, Ti Ren
spellingShingle	Huang, Ti Ren ddc 510 bkl 31.00 misc Extremal misc geodesic misc rigid curve misc Lie group Some properties of abnormal extremals on Lie groups
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topic_title	510 ASE 31.00 bkl Some properties of abnormal extremals on Lie groups Extremal (dpeaa)DE-He213 geodesic (dpeaa)DE-He213 rigid curve (dpeaa)DE-He213 Lie group (dpeaa)DE-He213
topic	ddc 510 bkl 31.00 misc Extremal misc geodesic misc rigid curve misc Lie group
topic_unstemmed	ddc 510 bkl 31.00 misc Extremal misc geodesic misc rigid curve misc Lie group
topic_browse	ddc 510 bkl 31.00 misc Extremal misc geodesic misc rigid curve misc Lie group
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title	Some properties of abnormal extremals on Lie groups
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title_full	Some properties of abnormal extremals on Lie groups
author_sort	Huang, Ti Ren
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title_sort	some properties of abnormal extremals on lie groups
title_auth	Some properties of abnormal extremals on Lie groups
abstract	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.
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.
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.
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title_short	Some properties of abnormal extremals on Lie groups
url	https://dx.doi.org/10.1007/s10114-014-1286-9
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doi_str	10.1007/s10114-014-1286-9
up_date	2024-07-04T00:20:26.755Z
_version_	1803605693439148032
fullrecord_marcxml	<?xml version="1.0" encoding="UTF-8"?><collection xmlns="http://www.loc.gov/MARC21/slim"><record><leader>01000caa a22002652 4500</leader><controlfield tag="001">SPR009018034</controlfield><controlfield tag="003">DE-627</controlfield><controlfield tag="005">20220110204927.0</controlfield><controlfield tag="007">cr uuu---uuuuu</controlfield><controlfield tag="008">201005s2014 xx \|\|\|\|\|o 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)SPR009018034</subfield></datafield><datafield tag="035" ind1=" " ind2=" "><subfield code="a">(SPR)s10114-014-1286-9-e</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">ASE</subfield></datafield><datafield tag="082" ind1="0" ind2="4"><subfield code="a">510</subfield><subfield code="q">ASE</subfield></datafield><datafield tag="084" ind1=" " ind2=" "><subfield code="a">31.00</subfield><subfield code="2">bkl</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">Computermedien</subfield><subfield code="b">c</subfield><subfield code="2">rdamedia</subfield></datafield><datafield tag="338" ind1=" " ind2=" "><subfield code="a">Online-Ressource</subfield><subfield code="b">cr</subfield><subfield code="2">rdacarrier</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><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">geodesic</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">rigid curve</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="650" ind1=" " ind2="4"><subfield code="a">Lie group</subfield><subfield code="7">(dpeaa)DE-He213</subfield></datafield><datafield tag="700" ind1="1" ind2=" "><subfield code="a">Yang, Xiao Ping</subfield><subfield code="e">verfasserin</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">Berlin : Springer, 1985</subfield><subfield code="g">30(2014), 12 vom: 15. 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