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001472390 019__ $$a1392164691$$a1392344311
001472390 020__ $$a9783031368578$$q(electronic bk.)
001472390 020__ $$a3031368576$$q(electronic bk.)
001472390 020__ $$z9783031368561
001472390 020__ $$z3031368568
001472390 0247_ $$a10.1007/978-3-031-36857-8$$2doi
001472390 035__ $$aSP(OCoLC)1393142110
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001472390 1001_ $$aGil-Medrano, O.$$q(Olga),$$eauthor.$$1https://orcid.org/0000-0002-0050-5960
001472390 24514 $$aThe volume of vector fields on Riemannian manifolds :$$bmain results and open problems /$$cOlga Gil-Medrano.
001472390 264_1 $$aCham :$$bSpringer,$$c2023.
001472390 300__ $$a1 online resource (viii, 126 pages).
001472390 336__ $$atext$$btxt$$2rdacontent
001472390 337__ $$acomputer$$bc$$2rdamedia
001472390 338__ $$aonline resource$$bcr$$2rdacarrier
001472390 4901_ $$aLecture notes in mathematics,$$x1617-9692 ;$$vvolume 2336
001472390 504__ $$aIncludes bibliographical references.
001472390 5050_ $$aIntro -- Preface -- Funding Acknowledgements -- Contents -- 1 Introduction -- 2 Minimal Sections of Tensor Bundles -- 2.1 Geometry of the Submanifold Determined by a Section of a Tensor Bundle -- 2.2 Minimal Sections of Tensor Bundles and Sphere Subbundles -- 2.3 First Variation of the Volume of Vector Fields: Minimal Vector Fields -- 2.4 Second Variation of the Volume of Vector Fields -- 2.5 The 2-Dimensional Case -- 2.6 Notes -- 2.6.1 Sections That Are Harmonic Maps -- 2.6.2 Sections That Are Critical Pointsof the Energy Functional -- 2.6.3 Minimal Oriented Distributions
001472390 5058_ $$a3 Minimal Vector Fields of Constant Length on the Odd-Dimensional Spheres -- 3.1 Minimality of the Hopf Vector Fields -- 3.2 Study of the Stability of the Hopf Vector Fields -- 3.3 Stability of the Hopf Vector Fields of Odd-Dimensional Space Forms of Positive Curvature -- 3.4 Notes -- 3.4.1 Spheres and Their Quotients with Berger Metrics -- 3.4.2 The Minimality Condition for Unit Killing Vector Fields -- 3.4.3 Minimality of the Characteristic Vector Field of a Contact Riemannian Manifold -- 3.4.4 Minimal Invariant Vector Fields on Lie Groups and Homogeneous Spaces
001472390 5058_ $$a3.4.5 Examples Related with Complex and Quaternionic Structures -- 4 Vector Fields of Constant Length of Minimum Volume on the Odd-Dimensional Spherical Space Forms -- 4.1 Hopf Vector Fields as Volume Minimisers in the 3-Dimensional Case -- 4.2 Hopf Vector Fields on 3-Dimensional Spheres with the Berger Metrics -- 4.3 Lower Bound of the Volume of Vector Fields of Constant Length -- 4.4 Asymptotic Behaviour of the Volume Functional -- 4.5 Notes -- 4.5.1 Unit Vector Fields on the Two-Dimensional Torus -- 4.5.2 Lower Bound of the Volume of Unit Vector Fields on Hypersurfaces of Rn+1
001472390 5058_ $$a4.5.3 Almost Hermitian Structures on S6 That Minimise the Volume -- 4.5.4 Minimisers of Functionals Related with the Energy -- 5 Vector Fields of Constant Length on Punctured Spheres -- 5.1 The Radial Vector Fields -- 5.2 Parallel Transport Vector Fields -- 5.3 The Main Open Problem -- 5.4 Area Minimising Vector Fields on the 2-Sphere -- 5.5 Notes -- 5.5.1 Radial Vector Fields on Riemannian Manifolds -- 5.5.2 Minimisers of the Volume Among Unit Vector Fields with Singular Points -- References
001472390 506__ $$aAccess limited to authorized users.
001472390 520__ $$aThis book focuses on the study of the volume of vector fields on Riemannian manifolds. Providing a thorough overview of research on vector fields defining minimal submanifolds, and on the existence and characterization of volume minimizers, it includes proofs of the most significant results obtained since the subject's introduction in 1986. Aiming to inspire further research, it also highlights a selection of intriguing open problems, and exhibits some previously unpublished results. The presentation is direct and deviates substantially from the usual approaches found in the literature, requiring a significant revision of definitions, statements, and proofs. A wide range of topics is covered, including: a discussion on the conditions for a vector field on a Riemannian manifold to determine a minimal submanifold within its tangent bundle with the Sasaki metric; numerous examples of minimal vector fields (including those of constant length on punctured spheres); a thorough analysis of Hopf vector fields on odd-dimensional spheres and their quotients; and a description of volume-minimizing vector fields of constant length on spherical space forms of dimension three. Each chapter concludes with an up-to-date survey which offers supplementary information and provides valuable insights into the material, enhancing the reader's understanding of the subject. Requiring a solid understanding of the fundamental concepts of Riemannian geometry, the book will be useful for researchers and PhD students with an interest in geometric analysis.
001472390 588__ $$aOnline resource; title from PDF title page (SpringerLink, viewed August 9, 2023).
001472390 650_0 $$aVector fields.
001472390 650_0 $$aRiemannian manifolds.
001472390 655_0 $$aElectronic books.
001472390 77608 $$iPrint version:$$aGil-Medrano, Olga$$tThe Volume of Vector Fields on Riemannian Manifolds$$dCham : Springer,c2023$$z9783031368561
001472390 830_0 $$aLecture notes in mathematics (Springer-Verlag) ;$$vv. 2336.$$x1617-9692
001472390 852__ $$bebk
001472390 85640 $$3Springer Nature$$uhttps://univsouthin.idm.oclc.org/login?url=https://link.springer.com/10.1007/978-3-031-36857-8$$zOnline Access$$91397441.1
001472390 909CO $$ooai:library.usi.edu:1472390$$pGLOBAL_SET
001472390 980__ $$aBIB
001472390 980__ $$aEBOOK
001472390 982__ $$aEbook
001472390 983__ $$aOnline
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