{"corpus_id":152112,"paper_sha":"ac939610c21d6b96458cb050bc9d216d1023d7c0","doi":null,"arxiv_id":"0704.2001","pmid":null,"pmcid":null,"mag_id":1651910053,"dblp_id":null,"acl_id":null,"title":"Geometry of Parallelizable Manifolds in the Context of Generalized Lagrange Spaces","year":2007,"publication_date":"2007-04-16","venue":"","journal":{"name":"arXiv: General Relativity and Quantum Cosmology","pages":null,"volume":""},"journal_issn":null,"journal_title":null,"publication_types":[],"pubmed_pub_types":null,"s2_fields_of_study":["Mathematics","Physics"],"reference_count":19,"citation_count":13,"influential_citation_count":0,"is_open_access":false,"arxiv_categories":["gr-qc","math.DG"],"arxiv_license":null,"arxiv_journal_ref":"Balkan J. Geom. Appl., 13,2 (2008), 120-139.","mesh_headings":null,"chemicals":null,"comments_corrections":null,"source_flags":1,"s2_open_access_pdf_url":null,"s2_open_access_landing_url":null,"s2_open_access_license":null,"s2_open_access_status":null,"pmc_open_access_pdf_url":null,"pmc_open_access_landing_url":null,"pmc_open_access_license":null,"pmc_open_access_status":null,"unpaywall_open_access_pdf_url":null,"unpaywall_open_access_landing_url":null,"unpaywall_open_access_license":null,"unpaywall_open_access_status":null,"abstract":"In this paper, we deal with a generalization of the geometry of parallelizable manifolds, or the absolute parallelism (AP-) geometry, in the context of generalized Lagrange spaces. All geometric objects de- flned in this geometry are not only functions of the positional argument x, but also depend on the directional argument y. In other words, instead of dealing with geometric objects deflned on the manifold M, as in the case of classical AP-geometry, we are dealing with geometric objects in the pullback bundle … i1 (TM) (the pullback of the tangent bundle TM by … : TM i! M). Many new geometric objects, which have no counterpart in the classical AP-geometry, emerge in this more general context. We refer to such a geometry as generalized AP-geometry (GAP-geometry). In analogy to AP-geometry, we deflne a d-connection in … i1 (TM) having remarkable properties, which we call the canonical d-connection, in terms of the unique torsion-free Riemannian d-connection. In addition to these two d-connections, two more d-connections are deflned, the dual and the symmetric d-connections. Our space, therefore, admits twelve curvature tensors (corresponding to the four deflned d-connections), three of which vanish identically. Simple formulae for the nine non-vanishing curvatures tensors are obtained, in terms of the torsion tensors of the canonical d- connection. The difierent W-tensors admitted by the space are also calcu- lated. All contractions of the h- and v-curvature tensors and the W-tensors are derived. Second rank symmetric and skew-symmetric tensors, which prove useful in physical applications, are singled out.","claims":[{"public_id":"cl_4aba00eb70fa3933b2ccb31c492d2527","status":"active","text":"A canonical d-connection is defined in terms of the unique torsion-free Riemannian d-connection.","confidence":0.95,"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_4aba00eb70fa3933b2ccb31c492d2527"},{"public_id":"cl_8db98b05c589c4b1d12b859dce95e2d4","status":"active","text":"Absolute parallelism geometry is generalized to generalized Lagrange spaces, yielding generalized AP-geometry in which geometric objects depend on both position x and direction y.","confidence":0.98,"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous 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identically.","confidence":0.97,"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_027c206df22ea546d51ee269b396e737"},{"public_id":"cl_5af1dd6a9c9522b55191f8082b0d60ae","status":"active","text":"The relevant geometric objects are formulated on the pullback bundle of the tangent bundle rather than only on the manifold.","confidence":0.93,"contributors":[{"id":1,"public_id":"12632b8b5f","public_label":"Anonymous (12632b8b5f)","roles":["extraction"],"url":"https://sah.borca.ai/u/12632b8b5f"}],"url":"https://sah.borca.ai/claims/cl_5af1dd6a9c9522b55191f8082b0d60ae"}],"concepts":[{"public_id":"co_06d520487956798f00754b669d507a41","status":"active","name":"curvature tensors","description":"Tensorial measures of curvature associated with the various d-connections in the 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