Volumes e curvaturas médias na geometria de Finsler:superfícies mínimas
| Autor(a) principal: | |
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| Data de Publicação: | 2012 |
| Tipo de documento: | Dissertação |
| Idioma: | por |
| Título da fonte: | Repositório Institucional da UFG |
| Texto Completo: | http://repositorio.bc.ufg.br/tede/handle/tde/2885 |
Resumo: | In Finsler geometry, we have several volume forms, hence various of mean curvature forms. The two best known volumes forms are the Busemann-Hausdorff and Holmes- Thompson volume form. The minimal surface with respect to these volume forms are called BH-minimal and HT-minimal surface, respectively. Let (R3; eFb) be a Minkowski space of Randers type with eFb = ea+eb; where ea is the Euclidean metric and eb = bdx3; 0 < b < 1: If a connected surface M in (R3; eFb) is minimal with respect to both volume forms Busemann-Hausdorff and Holmes-Thompson, then up to a parallel translation of R3; M is either a piece of plane or a piece of helicoid which is generated by lines screwing along the x3-axis. Furthermore it gives an explicit rotation hypersurfaces BH-minimal and HT-minimal generated by a plane curve around the axis in the direction of eb] in Minkowski (a;b)-space (Vn+1; eFb); where Vn+1 is an (n+1)-dimensional real vector space, eFb = eaf eb ea ; ea is the Euclidean metric, eb is a one form of constant length b = kebkea; eb] is the dual vector of eb with respect to ea: As an application, it give us an explicit expression of surface of rotation “ forward” BH-minimal generated by the rotation around the axis in the direction of eb] in Minkowski space of Randers type (V3; ea+eb): |
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Souza, Marcelo Almeida dehttp://lattes.cnpq.br/1343419041226215http://lattes.cnpq.br/5434324704126162Chavéz, Newton Mayer Solorzano2014-08-06T11:17:00Z2012-04-16CHAVES, Newton Mayer Solorzano. Volumes e curvaturas médias na geometria de Finsler: superfícies mínimas. 2012. 75 f. Dissertação (Mestrado em Matemática) - Universidade Federal de Goiás, Goiânia, 2012.http://repositorio.bc.ufg.br/tede/handle/tde/2885In Finsler geometry, we have several volume forms, hence various of mean curvature forms. The two best known volumes forms are the Busemann-Hausdorff and Holmes- Thompson volume form. The minimal surface with respect to these volume forms are called BH-minimal and HT-minimal surface, respectively. Let (R3; eFb) be a Minkowski space of Randers type with eFb = ea+eb; where ea is the Euclidean metric and eb = bdx3; 0 < b < 1: If a connected surface M in (R3; eFb) is minimal with respect to both volume forms Busemann-Hausdorff and Holmes-Thompson, then up to a parallel translation of R3; M is either a piece of plane or a piece of helicoid which is generated by lines screwing along the x3-axis. Furthermore it gives an explicit rotation hypersurfaces BH-minimal and HT-minimal generated by a plane curve around the axis in the direction of eb] in Minkowski (a;b)-space (Vn+1; eFb); where Vn+1 is an (n+1)-dimensional real vector space, eFb = eaf eb ea ; ea is the Euclidean metric, eb is a one form of constant length b = kebkea; eb] is the dual vector of eb with respect to ea: As an application, it give us an explicit expression of surface of rotation “ forward” BH-minimal generated by the rotation around the axis in the direction of eb] in Minkowski space of Randers type (V3; ea+eb):Na Geometria de Finsler, temos várias formas volume, consequentemente várias formas curvaturas médias. As duas mais conhecidas são as formas de volumes Busemann- Hausdorff e Holmes-Thompson. As superfícies mínimas com respeito a estes são chamados superfícies BH-mínimas e HT-mínimas, respectivamente. Seja (R3; eFb) um espaço de Minkowski do tipo Randers com eFb = ea+eb; onde ea é a métrica Euclidiana e eb = bdx3;0 < b < 1: Uma superfície em (R3; eFb) conexa M é mínima com respeito a ambas formas volumes Busemann-Hausdorff e Holmes-Thompson, então a menos de uma translação paralela de R3; M é parte de um plano ou parte de um helicóide, a qual é gerada pela rotação de uma reta (perpendicular ao eixo x3) ao longo do eixo x3: Ademais podemos obter explicitamente hipersuperfícies de rotação BH-mínima e HT-mínima geradas por uma curva plana em torno do eixo na direção de eb] num espaço (a; b) de Minkowski (Vn+1; eFb); onde Vn+1 é um espaço vetorial de dimensão (n+1); eFb = eaf eb ea ; ea é a métrica Euclidiana, eb é uma 1-forma constante com norma b := kebkea; eb] é o vetor dual de eb com respeito a a: Como aplicação, se dá uma expressão explícita de superfície de rotação completa “forward” BH-mínima gerada pela rotação em torno do eixo na direção de eb] num espaço de Minkowski do tipo Randers (V3; ea+eb):Submitted by Cássia Santos (cassia.bcufg@gmail.com) on 2014-08-06T11:17:00Z No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Volumes_e_curvaturas_medias_na_geometria_de_finsler.pdf: 818570 bytes, checksum: fce77ff7f92ae9cc2bf9af2aa0318c4c (MD5)Made available in DSpace on 2014-08-06T11:17:00Z (GMT). No. of bitstreams: 2 license_rdf: 23148 bytes, checksum: 9da0b6dfac957114c6a7714714b86306 (MD5) Volumes_e_curvaturas_medias_na_geometria_de_finsler.pdf: 818570 bytes, checksum: fce77ff7f92ae9cc2bf9af2aa0318c4c (MD5) Previous issue date: 2012-04-16application/pdfhttp://repositorio.bc.ufg.br/tede/retrieve/6062/Volumes_e_curvaturas_medias_na_geometria_de_finsler.pdf.jpgporUniversidade Federal de GoiásPrograma de Pós-graduação em Matemática (IME)UFGBrasilInstituto de Matemática e Estatística - IME (RG)[1] Bao, D.; Chern, S.S., Shen,Z., An Introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics 200, Springer-Verlag, New York, Inc., (2000). [2] Cherg, X., Shen, Z, A Class of Finsler Metrics with Isotropic S-Curvature, Israel J. Math. 169, 317-340 (2009). [3] Cui, N., Shen, Y.B., Bernstein Type Theorems for Minimal Surfaces in (a;b)-space Public Math. Debrecen 74, 383-400 (2009). [4] Cui, N., Shen, Y.B., Minimal Rotational Hypersurfaces in Minkowski (a;b)-space, Geom Dedicata. (2010) [5] do Carmo, M.P., Differential Geometry of Curves and Surfaces, Prentice-Hall, New Jersey, (1976). [6] do Carmo, M. P., Formas Diferenciáveis e Aplicações, Monografias de Matemática n 37, 8vo. Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro, (1983). [7] do Carmo, M.P., Geometria Riemanniana, Projeto Euclides, IMPA, Rio de Janeiro, 1988, 2 edição. [8] do Carmo, M.P., O Método do Referencial Móvel, Escola Latino-Americana de Matemática, IMPA, Rio de Janeiro, (1976). [9] D.Bao, C.Robles and Z.Shen, Zermelo Navigation on Riemannian Manifolds, J. Diferential Geom. 66 (2004), no. 3, 377-435. [10] He, Q., Shen, Y.B., On Bernstein type theorems in Finsler geometry with volume form induced from sphere bundle, proc. Amer. Math. Soc. 134, 871-880 (2006). [11] He, Q., Shen, Y.B., On the Mean Curvature of Finsler Submanifolds, Chinese J. Contemp. 27A, 663-674 (2006). [12] Lima, E. L., Álgebra Linear, Coleção Matemática Universitária, IMPA, Rio de Janeiro, 2000, 4 edição. [13] Lima, E. L., Análise no Espaço Rn, Coleção Matemática Universitária, IMPA, Rio de Janeiro, (2007). [14] Shen, Z.; Chern, S.S.; Riemann Finsler Geometry, World Scientific, Nankai Tracts in Mathematics, Volume 6. [15] Shen, Z., Lectures on Finsler Geometry, World Sci., Singapore (2001). [16] Shen, Z., On Finsler geometry of submanifolds., Math. Ann. 131 549-576 (1998). [17] Souza, M., Spruck, J., Tenenblat, K.,A Bernstein type theorem on a Randers space, Math. Ann. 329, 291-305 (2004). [18] Souza, M., Tenenblat, K., Minimal Surfaces of Rotation in Finsler Space with a Randers Metric, Math. Ann. 325, 625-642 (2003). [19] S.S. Chern, Z. Shen, Riemann-Finsler Geometry, Nankai Tracts in Mathematics, vol. 6, World Scientific (2005). [20] Tenenblat, K., Introdução à Geometria Diferencial, Editora da Unb, Brasília, (1998). [21] Wu, B.Y., On the Folume Forms and Submanifolds in Finsler Geometry. Chinese. J. Contemp. Math. 27, 61-72 (2006). [22] Wu, B.Y., A Local Rigidity Theorem for Minimal Surfaces in Minkowski 3-space of Randers type., Ann. Glob. Anal. Geom. 31, 375-384 (2007).6600717948137941247600600600-4268777512335152015-7090823417984401694http://creativecommons.org/licenses/by-nc-nd/4.0/info:eu-repo/semantics/openAccessEspaços de Minkowski do tipo RandersCurvatura MédiaBH-mínimaHT-mínimaMinkowski Space of Randers typeMean CurvatureBH-minimalHT-minimalCIENCIAS EXATAS E DA TERRA::MATEMATICAVolumes e curvaturas médias na geometria de Finsler:superfícies mínimasVolumes and means curvatures in Finsler geometry: minimal surfacesinfo:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/masterThesisreponame:Repositório Institucional da UFGinstname:Universidade Federal de Goiás (UFG)instacron:UFGLICENSElicense.txtlicense.txttext/plain; charset=utf-82142http://repositorio.bc.ufg.br/tede/bitstreams/e62865ba-91c4-4765-b760-44936ede39e6/download232e528055260031f4e2af4136033daaMD51CC-LICENSElicense_urllicense_urltext/plain; charset=utf-849http://repositorio.bc.ufg.br/tede/bitstreams/722156f1-5c5a-4450-89b8-43ecc804cb50/download4afdbb8c545fd630ea7db775da747b2fMD52license_textlicense_texttext/html; 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| dc.title.por.fl_str_mv |
Volumes e curvaturas médias na geometria de Finsler:superfícies mínimas |
| dc.title.alternative.eng.fl_str_mv |
Volumes and means curvatures in Finsler geometry: minimal surfaces |
| title |
Volumes e curvaturas médias na geometria de Finsler:superfícies mínimas |
| spellingShingle |
Volumes e curvaturas médias na geometria de Finsler:superfícies mínimas Chavéz, Newton Mayer Solorzano Espaços de Minkowski do tipo Randers Curvatura Média BH-mínima HT-mínima Minkowski Space of Randers type Mean Curvature BH-minimal HT-minimal CIENCIAS EXATAS E DA TERRA::MATEMATICA |
| title_short |
Volumes e curvaturas médias na geometria de Finsler:superfícies mínimas |
| title_full |
Volumes e curvaturas médias na geometria de Finsler:superfícies mínimas |
| title_fullStr |
Volumes e curvaturas médias na geometria de Finsler:superfícies mínimas |
| title_full_unstemmed |
Volumes e curvaturas médias na geometria de Finsler:superfícies mínimas |
| title_sort |
Volumes e curvaturas médias na geometria de Finsler:superfícies mínimas |
| author |
Chavéz, Newton Mayer Solorzano |
| author_facet |
Chavéz, Newton Mayer Solorzano |
| author_role |
author |
| dc.contributor.advisor1.fl_str_mv |
Souza, Marcelo Almeida de |
| dc.contributor.advisor1Lattes.fl_str_mv |
http://lattes.cnpq.br/1343419041226215 |
| dc.contributor.authorLattes.fl_str_mv |
http://lattes.cnpq.br/5434324704126162 |
| dc.contributor.author.fl_str_mv |
Chavéz, Newton Mayer Solorzano |
| contributor_str_mv |
Souza, Marcelo Almeida de |
| dc.subject.por.fl_str_mv |
Espaços de Minkowski do tipo Randers Curvatura Média BH-mínima HT-mínima |
| topic |
Espaços de Minkowski do tipo Randers Curvatura Média BH-mínima HT-mínima Minkowski Space of Randers type Mean Curvature BH-minimal HT-minimal CIENCIAS EXATAS E DA TERRA::MATEMATICA |
| dc.subject.eng.fl_str_mv |
Minkowski Space of Randers type Mean Curvature BH-minimal HT-minimal |
| dc.subject.cnpq.fl_str_mv |
CIENCIAS EXATAS E DA TERRA::MATEMATICA |
| description |
In Finsler geometry, we have several volume forms, hence various of mean curvature forms. The two best known volumes forms are the Busemann-Hausdorff and Holmes- Thompson volume form. The minimal surface with respect to these volume forms are called BH-minimal and HT-minimal surface, respectively. Let (R3; eFb) be a Minkowski space of Randers type with eFb = ea+eb; where ea is the Euclidean metric and eb = bdx3; 0 < b < 1: If a connected surface M in (R3; eFb) is minimal with respect to both volume forms Busemann-Hausdorff and Holmes-Thompson, then up to a parallel translation of R3; M is either a piece of plane or a piece of helicoid which is generated by lines screwing along the x3-axis. Furthermore it gives an explicit rotation hypersurfaces BH-minimal and HT-minimal generated by a plane curve around the axis in the direction of eb] in Minkowski (a;b)-space (Vn+1; eFb); where Vn+1 is an (n+1)-dimensional real vector space, eFb = eaf eb ea ; ea is the Euclidean metric, eb is a one form of constant length b = kebkea; eb] is the dual vector of eb with respect to ea: As an application, it give us an explicit expression of surface of rotation “ forward” BH-minimal generated by the rotation around the axis in the direction of eb] in Minkowski space of Randers type (V3; ea+eb): |
| publishDate |
2012 |
| dc.date.issued.fl_str_mv |
2012-04-16 |
| dc.date.accessioned.fl_str_mv |
2014-08-06T11:17:00Z |
| dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
| dc.type.driver.fl_str_mv |
info:eu-repo/semantics/masterThesis |
| format |
masterThesis |
| status_str |
publishedVersion |
| dc.identifier.citation.fl_str_mv |
CHAVES, Newton Mayer Solorzano. Volumes e curvaturas médias na geometria de Finsler: superfícies mínimas. 2012. 75 f. Dissertação (Mestrado em Matemática) - Universidade Federal de Goiás, Goiânia, 2012. |
| dc.identifier.uri.fl_str_mv |
http://repositorio.bc.ufg.br/tede/handle/tde/2885 |
| identifier_str_mv |
CHAVES, Newton Mayer Solorzano. Volumes e curvaturas médias na geometria de Finsler: superfícies mínimas. 2012. 75 f. Dissertação (Mestrado em Matemática) - Universidade Federal de Goiás, Goiânia, 2012. |
| url |
http://repositorio.bc.ufg.br/tede/handle/tde/2885 |
| dc.language.iso.fl_str_mv |
por |
| language |
por |
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6600717948137941247 |
| dc.relation.confidence.fl_str_mv |
600 600 600 |
| dc.relation.department.fl_str_mv |
-4268777512335152015 |
| dc.relation.cnpq.fl_str_mv |
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[1] Bao, D.; Chern, S.S., Shen,Z., An Introduction to Riemann-Finsler Geometry, Graduate Texts in Mathematics 200, Springer-Verlag, New York, Inc., (2000). [2] Cherg, X., Shen, Z, A Class of Finsler Metrics with Isotropic S-Curvature, Israel J. Math. 169, 317-340 (2009). [3] Cui, N., Shen, Y.B., Bernstein Type Theorems for Minimal Surfaces in (a;b)-space Public Math. Debrecen 74, 383-400 (2009). [4] Cui, N., Shen, Y.B., Minimal Rotational Hypersurfaces in Minkowski (a;b)-space, Geom Dedicata. (2010) [5] do Carmo, M.P., Differential Geometry of Curves and Surfaces, Prentice-Hall, New Jersey, (1976). [6] do Carmo, M. P., Formas Diferenciáveis e Aplicações, Monografias de Matemática n 37, 8vo. Colóquio Brasileiro de Matemática, IMPA, Rio de Janeiro, (1983). [7] do Carmo, M.P., Geometria Riemanniana, Projeto Euclides, IMPA, Rio de Janeiro, 1988, 2 edição. [8] do Carmo, M.P., O Método do Referencial Móvel, Escola Latino-Americana de Matemática, IMPA, Rio de Janeiro, (1976). [9] D.Bao, C.Robles and Z.Shen, Zermelo Navigation on Riemannian Manifolds, J. Diferential Geom. 66 (2004), no. 3, 377-435. [10] He, Q., Shen, Y.B., On Bernstein type theorems in Finsler geometry with volume form induced from sphere bundle, proc. Amer. Math. Soc. 134, 871-880 (2006). [11] He, Q., Shen, Y.B., On the Mean Curvature of Finsler Submanifolds, Chinese J. Contemp. 27A, 663-674 (2006). [12] Lima, E. L., Álgebra Linear, Coleção Matemática Universitária, IMPA, Rio de Janeiro, 2000, 4 edição. [13] Lima, E. L., Análise no Espaço Rn, Coleção Matemática Universitária, IMPA, Rio de Janeiro, (2007). [14] Shen, Z.; Chern, S.S.; Riemann Finsler Geometry, World Scientific, Nankai Tracts in Mathematics, Volume 6. [15] Shen, Z., Lectures on Finsler Geometry, World Sci., Singapore (2001). [16] Shen, Z., On Finsler geometry of submanifolds., Math. Ann. 131 549-576 (1998). [17] Souza, M., Spruck, J., Tenenblat, K.,A Bernstein type theorem on a Randers space, Math. Ann. 329, 291-305 (2004). [18] Souza, M., Tenenblat, K., Minimal Surfaces of Rotation in Finsler Space with a Randers Metric, Math. Ann. 325, 625-642 (2003). [19] S.S. Chern, Z. Shen, Riemann-Finsler Geometry, Nankai Tracts in Mathematics, vol. 6, World Scientific (2005). [20] Tenenblat, K., Introdução à Geometria Diferencial, Editora da Unb, Brasília, (1998). [21] Wu, B.Y., On the Folume Forms and Submanifolds in Finsler Geometry. Chinese. J. Contemp. Math. 27, 61-72 (2006). [22] Wu, B.Y., A Local Rigidity Theorem for Minimal Surfaces in Minkowski 3-space of Randers type., Ann. Glob. Anal. Geom. 31, 375-384 (2007). |
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