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020 _a9783031024160
_9978-3-031-02416-0
024 7 _a10.1007/978-3-031-02416-0
_2doi
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072 7 _aPB
_2bicssc
072 7 _aMAT000000
_2bisacsh
072 7 _aPB
_2thema
082 0 4 _a510
_223
100 1 _aCalviño-Louzao, Esteban.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_981241
245 1 0 _aAspects of Differential Geometry IV
_h[electronic resource] /
_cby Esteban Calviño-Louzao, Eduardo García-Río, Peter Gilkey, JeongHyeong Park, Ramón Vázquez-Lorenzo.
250 _a1st ed. 2019.
264 1 _aCham :
_bSpringer International Publishing :
_bImprint: Springer,
_c2019.
300 _aXVII, 149 p.
_bonline resource.
336 _atext
_btxt
_2rdacontent
337 _acomputer
_bc
_2rdamedia
338 _aonline resource
_bcr
_2rdacarrier
347 _atext file
_bPDF
_2rda
490 1 _aSynthesis Lectures on Mathematics & Statistics,
_x1938-1751
505 0 _aPreface -- Acknowledgments -- An Introduction to Affine Geometry -- The Geometry of Type A Models -- The Geometry of Type B Models -- Applications of Affine Surface Theory -- Bibliography -- Authors' Biographies -- Index .
520 _aBook IV continues the discussion begun in the first three volumes. Although it is aimed at first-year graduate students, it is also intended to serve as a basic reference for people working in affine differential geometry. It also should be accessible to undergraduates interested in affine differential geometry. We are primarily concerned with the study of affine surfaces {which} are locally homogeneous. We discuss affine gradient Ricci solitons, affine Killing vector fields, and geodesic completeness. Opozda has classified the affine surface geometries which are locally homogeneous; we follow her classification. Up to isomorphism, there are two simply connected Lie groups of dimension 2. The translation group ℝ² is Abelian and the �������� + ���� group\index{ax+b group} is non-Abelian. The first chapter presents foundational material. The second chapter deals with Type ���� surfaces. These are the left-invariant affine geometries on ℝ². Associating to each Type ���� surface the space of solutions to the quasi-Einstein equation corresponding to the eigenvalue ����=-1$ turns out to be a very powerful technique and plays a central role in our study as it links an analytic invariant with the underlying geometry of the surface. The third chapter deals with Type ���� surfaces; these are the left-invariant affine geometries on the �������� + ���� group. These geometries form a very rich family which is only partially understood. The only remaining homogeneous geometry is that of the sphere ����². The fourth chapter presents relations between the geometry of an affine surface and the geometry of the cotangent bundle equipped with the neutral signature metric of the modified Riemannian extension.
650 0 _aMathematics.
_911584
650 0 _aStatistics .
_931616
650 0 _aEngineering mathematics.
_93254
650 1 4 _aMathematics.
_911584
650 2 4 _aStatistics.
_914134
650 2 4 _aEngineering Mathematics.
_93254
700 1 _aGarcía-Río, Eduardo.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_981242
700 1 _aGilkey, Peter.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_981243
700 1 _aPark, JeongHyeong.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_981244
700 1 _aVázquez-Lorenzo, Ramón.
_eauthor.
_4aut
_4http://id.loc.gov/vocabulary/relators/aut
_981245
710 2 _aSpringerLink (Online service)
_981246
773 0 _tSpringer Nature eBook
776 0 8 _iPrinted edition:
_z9783031002625
776 0 8 _iPrinted edition:
_z9783031012884
776 0 8 _iPrinted edition:
_z9783031035449
830 0 _aSynthesis Lectures on Mathematics & Statistics,
_x1938-1751
_981247
856 4 0 _uhttps://doi.org/10.1007/978-3-031-02416-0
912 _aZDB-2-SXSC
942 _cEBK
999 _c85137
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