On the volume of unit vector fields on spaces of constant sectional curvature

Abstract A unit vector field X on a Riemannian manifold determines a submanifold in the unit tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. It is known that the parallel fields are the trivial minima. In this paper, we obtain a lower bound for the vo...
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Autor*in:	Brito, Fabiano B. [verfasserIn] Chacón, Pablo M. Naveira, A. M.

Format:	Artikel
Sprache:	Englisch

Erschienen:	2004

Systematik:	SA 3400 SA 3400

Anmerkung:	© Birkhäuser-Verlag 2004

Übergeordnetes Werk:	Enthalten in: Commentarii mathematici Helvetici - Springer International Publishing, 1929, 79(2004), 2 vom: 01. Apr., Seite 300-316
Übergeordnetes Werk:	volume:79 ; year:2004 ; number:2 ; day:01 ; month:04 ; pages:300-316

Links:	Volltext

DOI / URN:	10.1007/s00014-004-0802-4

Katalog-ID:	OLC2069326349

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520			\|a Abstract A unit vector field X on a Riemannian manifold determines a submanifold in the unit tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. It is known that the parallel fields are the trivial minima. In this paper, we obtain a lower bound for the volume in terms of the integrals of the 2i-symmetric functions of the second fundamental form of the orthogonal distribution to the field X. In the spheres ${\textbf {S}}^{2k+1}$, this lower bound is independent of X. Consequently, the volume of a unit vector field on an odd-sphere is always greater than the volume of the radial field. The main theorem on volumes is applied also to hyperbolic compact spaces, giving a non-trivial lower bound of the volume of unit fields.
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allfields	10.1007/s00014-004-0802-4 doi (DE-627)OLC2069326349 (DE-He213)s00014-004-0802-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn SA 3400 VZ rvk SA 3400 VZ rvk Brito, Fabiano B. verfasserin aut On the volume of unit vector fields on spaces of constant sectional curvature 2004 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Birkhäuser-Verlag 2004 Abstract A unit vector field X on a Riemannian manifold determines a submanifold in the unit tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. It is known that the parallel fields are the trivial minima. In this paper, we obtain a lower bound for the volume in terms of the integrals of the 2i-symmetric functions of the second fundamental form of the orthogonal distribution to the field X. In the spheres ${\textbf {S}}^{2k+1}$, this lower bound is independent of X. Consequently, the volume of a unit vector field on an odd-sphere is always greater than the volume of the radial field. The main theorem on volumes is applied also to hyperbolic compact spaces, giving a non-trivial lower bound of the volume of unit fields. Chacón, Pablo M. aut Naveira, A. M. aut Enthalten in Commentarii mathematici Helvetici Springer International Publishing, 1929 79(2004), 2 vom: 01. Apr., Seite 300-316 (DE-627)129068411 (DE-600)1555-6 (DE-576)014399989 0010-2571 nnns volume:79 year:2004 number:2 day:01 month:04 pages:300-316 https://doi.org/10.1007/s00014-004-0802-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_24 GBV_ILN_40 GBV_ILN_65 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2007 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4029 GBV_ILN_4036 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4318 GBV_ILN_4323 GBV_ILN_4325 SA 3400 SA 3400 AR 79 2004 2 01 04 300-316
spelling	10.1007/s00014-004-0802-4 doi (DE-627)OLC2069326349 (DE-He213)s00014-004-0802-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn SA 3400 VZ rvk SA 3400 VZ rvk Brito, Fabiano B. verfasserin aut On the volume of unit vector fields on spaces of constant sectional curvature 2004 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Birkhäuser-Verlag 2004 Abstract A unit vector field X on a Riemannian manifold determines a submanifold in the unit tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. It is known that the parallel fields are the trivial minima. In this paper, we obtain a lower bound for the volume in terms of the integrals of the 2i-symmetric functions of the second fundamental form of the orthogonal distribution to the field X. In the spheres ${\textbf {S}}^{2k+1}$, this lower bound is independent of X. Consequently, the volume of a unit vector field on an odd-sphere is always greater than the volume of the radial field. The main theorem on volumes is applied also to hyperbolic compact spaces, giving a non-trivial lower bound of the volume of unit fields. Chacón, Pablo M. aut Naveira, A. M. aut Enthalten in Commentarii mathematici Helvetici Springer International Publishing, 1929 79(2004), 2 vom: 01. Apr., Seite 300-316 (DE-627)129068411 (DE-600)1555-6 (DE-576)014399989 0010-2571 nnns volume:79 year:2004 number:2 day:01 month:04 pages:300-316 https://doi.org/10.1007/s00014-004-0802-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_24 GBV_ILN_40 GBV_ILN_65 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2007 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4029 GBV_ILN_4036 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4318 GBV_ILN_4323 GBV_ILN_4325 SA 3400 SA 3400 AR 79 2004 2 01 04 300-316
allfields_unstemmed	10.1007/s00014-004-0802-4 doi (DE-627)OLC2069326349 (DE-He213)s00014-004-0802-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn SA 3400 VZ rvk SA 3400 VZ rvk Brito, Fabiano B. verfasserin aut On the volume of unit vector fields on spaces of constant sectional curvature 2004 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Birkhäuser-Verlag 2004 Abstract A unit vector field X on a Riemannian manifold determines a submanifold in the unit tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. It is known that the parallel fields are the trivial minima. In this paper, we obtain a lower bound for the volume in terms of the integrals of the 2i-symmetric functions of the second fundamental form of the orthogonal distribution to the field X. In the spheres ${\textbf {S}}^{2k+1}$, this lower bound is independent of X. Consequently, the volume of a unit vector field on an odd-sphere is always greater than the volume of the radial field. The main theorem on volumes is applied also to hyperbolic compact spaces, giving a non-trivial lower bound of the volume of unit fields. Chacón, Pablo M. aut Naveira, A. M. aut Enthalten in Commentarii mathematici Helvetici Springer International Publishing, 1929 79(2004), 2 vom: 01. Apr., Seite 300-316 (DE-627)129068411 (DE-600)1555-6 (DE-576)014399989 0010-2571 nnns volume:79 year:2004 number:2 day:01 month:04 pages:300-316 https://doi.org/10.1007/s00014-004-0802-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_24 GBV_ILN_40 GBV_ILN_65 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2007 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4029 GBV_ILN_4036 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4318 GBV_ILN_4323 GBV_ILN_4325 SA 3400 SA 3400 AR 79 2004 2 01 04 300-316
allfieldsGer	10.1007/s00014-004-0802-4 doi (DE-627)OLC2069326349 (DE-He213)s00014-004-0802-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn SA 3400 VZ rvk SA 3400 VZ rvk Brito, Fabiano B. verfasserin aut On the volume of unit vector fields on spaces of constant sectional curvature 2004 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Birkhäuser-Verlag 2004 Abstract A unit vector field X on a Riemannian manifold determines a submanifold in the unit tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. It is known that the parallel fields are the trivial minima. In this paper, we obtain a lower bound for the volume in terms of the integrals of the 2i-symmetric functions of the second fundamental form of the orthogonal distribution to the field X. In the spheres ${\textbf {S}}^{2k+1}$, this lower bound is independent of X. Consequently, the volume of a unit vector field on an odd-sphere is always greater than the volume of the radial field. The main theorem on volumes is applied also to hyperbolic compact spaces, giving a non-trivial lower bound of the volume of unit fields. Chacón, Pablo M. aut Naveira, A. M. aut Enthalten in Commentarii mathematici Helvetici Springer International Publishing, 1929 79(2004), 2 vom: 01. Apr., Seite 300-316 (DE-627)129068411 (DE-600)1555-6 (DE-576)014399989 0010-2571 nnns volume:79 year:2004 number:2 day:01 month:04 pages:300-316 https://doi.org/10.1007/s00014-004-0802-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_24 GBV_ILN_40 GBV_ILN_65 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2007 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4029 GBV_ILN_4036 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4318 GBV_ILN_4323 GBV_ILN_4325 SA 3400 SA 3400 AR 79 2004 2 01 04 300-316
allfieldsSound	10.1007/s00014-004-0802-4 doi (DE-627)OLC2069326349 (DE-He213)s00014-004-0802-4-p DE-627 ger DE-627 rakwb eng 510 VZ 17,1 ssgn SA 3400 VZ rvk SA 3400 VZ rvk Brito, Fabiano B. verfasserin aut On the volume of unit vector fields on spaces of constant sectional curvature 2004 Text txt rdacontent ohne Hilfsmittel zu benutzen n rdamedia Band nc rdacarrier © Birkhäuser-Verlag 2004 Abstract A unit vector field X on a Riemannian manifold determines a submanifold in the unit tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. It is known that the parallel fields are the trivial minima. In this paper, we obtain a lower bound for the volume in terms of the integrals of the 2i-symmetric functions of the second fundamental form of the orthogonal distribution to the field X. In the spheres ${\textbf {S}}^{2k+1}$, this lower bound is independent of X. Consequently, the volume of a unit vector field on an odd-sphere is always greater than the volume of the radial field. The main theorem on volumes is applied also to hyperbolic compact spaces, giving a non-trivial lower bound of the volume of unit fields. Chacón, Pablo M. aut Naveira, A. M. aut Enthalten in Commentarii mathematici Helvetici Springer International Publishing, 1929 79(2004), 2 vom: 01. Apr., Seite 300-316 (DE-627)129068411 (DE-600)1555-6 (DE-576)014399989 0010-2571 nnns volume:79 year:2004 number:2 day:01 month:04 pages:300-316 https://doi.org/10.1007/s00014-004-0802-4 lizenzpflichtig Volltext GBV_USEFLAG_A SYSFLAG_A GBV_OLC SSG-OLC-MAT SSG-OPC-MAT GBV_ILN_11 GBV_ILN_20 GBV_ILN_22 GBV_ILN_24 GBV_ILN_40 GBV_ILN_65 GBV_ILN_70 GBV_ILN_2002 GBV_ILN_2004 GBV_ILN_2005 GBV_ILN_2007 GBV_ILN_2010 GBV_ILN_2012 GBV_ILN_2088 GBV_ILN_2409 GBV_ILN_4027 GBV_ILN_4029 GBV_ILN_4036 GBV_ILN_4125 GBV_ILN_4126 GBV_ILN_4277 GBV_ILN_4305 GBV_ILN_4306 GBV_ILN_4307 GBV_ILN_4313 GBV_ILN_4318 GBV_ILN_4323 GBV_ILN_4325 SA 3400 SA 3400 AR 79 2004 2 01 04 300-316
language	English
source	Enthalten in Commentarii mathematici Helvetici 79(2004), 2 vom: 01. Apr., Seite 300-316 volume:79 year:2004 number:2 day:01 month:04 pages:300-316
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author	Brito, Fabiano B.
spellingShingle	Brito, Fabiano B. ddc 510 ssgn 17,1 rvk SA 3400 On the volume of unit vector fields on spaces of constant sectional curvature
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topic_title	510 VZ 17,1 ssgn SA 3400 VZ rvk On the volume of unit vector fields on spaces of constant sectional curvature
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title	On the volume of unit vector fields on spaces of constant sectional curvature
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title_full	On the volume of unit vector fields on spaces of constant sectional curvature
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title_sort	on the volume of unit vector fields on spaces of constant sectional curvature
title_auth	On the volume of unit vector fields on spaces of constant sectional curvature
abstract	Abstract A unit vector field X on a Riemannian manifold determines a submanifold in the unit tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. It is known that the parallel fields are the trivial minima. In this paper, we obtain a lower bound for the volume in terms of the integrals of the 2i-symmetric functions of the second fundamental form of the orthogonal distribution to the field X. In the spheres ${\textbf {S}}^{2k+1}$, this lower bound is independent of X. Consequently, the volume of a unit vector field on an odd-sphere is always greater than the volume of the radial field. The main theorem on volumes is applied also to hyperbolic compact spaces, giving a non-trivial lower bound of the volume of unit fields. © Birkhäuser-Verlag 2004
abstractGer	Abstract A unit vector field X on a Riemannian manifold determines a submanifold in the unit tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. It is known that the parallel fields are the trivial minima. In this paper, we obtain a lower bound for the volume in terms of the integrals of the 2i-symmetric functions of the second fundamental form of the orthogonal distribution to the field X. In the spheres ${\textbf {S}}^{2k+1}$, this lower bound is independent of X. Consequently, the volume of a unit vector field on an odd-sphere is always greater than the volume of the radial field. The main theorem on volumes is applied also to hyperbolic compact spaces, giving a non-trivial lower bound of the volume of unit fields. © Birkhäuser-Verlag 2004
abstract_unstemmed	Abstract A unit vector field X on a Riemannian manifold determines a submanifold in the unit tangent bundle. The volume of X is the volume of this submanifold for the induced Sasaki metric. It is known that the parallel fields are the trivial minima. In this paper, we obtain a lower bound for the volume in terms of the integrals of the 2i-symmetric functions of the second fundamental form of the orthogonal distribution to the field X. In the spheres ${\textbf {S}}^{2k+1}$, this lower bound is independent of X. Consequently, the volume of a unit vector field on an odd-sphere is always greater than the volume of the radial field. The main theorem on volumes is applied also to hyperbolic compact spaces, giving a non-trivial lower bound of the volume of unit fields. © Birkhäuser-Verlag 2004
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title_short	On the volume of unit vector fields on spaces of constant sectional curvature
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