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The Prescribed Curvature Problem in Low Dimension

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Geometry, Algebra and Applications: From Mechanics to Cryptography

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 161))

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Abstract

We describe some recent results concerning the inverse curvature problem, that is, the existence and description of metrics with prescribed curvature, focusing on the low-dimensional homogeneous cases.

Dedicated to Jaime Muñoz Masqué on the occasion of his 65th birthday

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References

  1. Calvaruso, G.: Homogeneous structures on three-dimensional Lorentzian manifolds. J. Geom. Phys. 57, 1279–1291 (2007); Addendum: J. Geom. Phys. 58, 291–292 (2008)

    Google Scholar 

  2. Calvaruso, G.: Einstein-like metrics on three-dimensional homogeneous Lorentzian manifolds. Geometriae Dedicata 127, 99–119 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  3. Calvaruso, G.: Three-dimensional homogeneous Lorentzian metrics with prescribed Ricci tensor. J. Math. Phys. 48, 123518, 17 p. (2007)

    Google Scholar 

  4. Calvaruso, G.: Pseudo-Riemannian \(3\)-manifolds with prescribed distinct constant Ricci eigenvalues. Differ. Geom. Appl. 26, 419–433 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  5. Calvaruso, G.: Curvature homogeneous Lorentzian three-manifolds. Ann. Glob. Anal. Geom. 36, 1–17 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  6. DeTurck, D.M.: The equation of prescribed Ricci curvature. Bull. Am. Math. Soc. 3, 701–704 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  7. DeTurck, D.M.: Existence of metrics with prescribed Ricci curvature: local theory. Invent. Math. 65, 179–207 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  8. DeTurck, D.M.: The Cauchy problem for Lorentz metrics with prescribed Ricci curvature. Compos. Math. 48, 327–349 (1983)

    MathSciNet  MATH  Google Scholar 

  9. DeTurck, D.M., Goldschmidt, H., Talvacchia, J.: Connections with prescribed curvature and Yang-Mills currents: the semi-simple case. Ann. Sci. Ecole Norm. S. 24, 57–112 (1991)

    MathSciNet  MATH  Google Scholar 

  10. DeTurck, D.M., Goldschmidt, H., Talvacchia, J.: Local existence of connections with prescribed curvature. In: Differential Geometry, Global Analysis, and Topology. Halifax, NS (1990), pp. 13-25. In: CMS Conf. Proc., vol. 12, Amer. Math. Soc., Providence, RI (1991)

    Google Scholar 

  11. Muñoz-Masqué, J., Pozo Coronado, L.M., Sánchez Rodríguez, I. J.: The prescribed curvature problem in dimension four. Math. Pure Appl. 92, 599–612 (2009)

    Google Scholar 

  12. O’Neill, B.: Semi-Riemannian Geometry. Academic Press, New York (1983)

    MATH  Google Scholar 

  13. Sekigawa, K.: On some three-dimensional curvature homogeneous spaces. Tensor 31, 87–97 (1977)

    MathSciNet  MATH  Google Scholar 

  14. Tsarev, S.P.: Which \(2\)-forms are locally curvature forms? Funct. Anal. Appl. 16, 235–237 (1982)

    Article  MathSciNet  MATH  Google Scholar 

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Acknowledgments

Partially supported by the Ministry of Economy and Competitiveness, Spain, project MTM2011-22528.

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Authors and Affiliations

  1. Department of Mathematics and Physics “Ennio De Giorgi”, University of Salento, 73100, Lecce, LE, Italy

    Giovanni Calvaruso

Authors
  1. Giovanni Calvaruso
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Correspondence to Giovanni Calvaruso .

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Editors and Affiliations

  1. Departamento de Geometría y Topología, Universidad Complutense de Madrid, Madrid, Spain

    Marco Castrillón López

  2. Physical and Information Technology, Spanish National Research Council (CSIC), Madrid, Spain

    Luis Hernández Encinas

  3. Institute of Fundamental Physics, Spanish National Research Council (CSIC), Madrid, Spain

    Pedro Martínez Gadea

  4. Departamento de Matemática Aplicada, Universidad Politécnica de Madrid, Madrid, Spain

    Mª Eugenia Rosado María

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Calvaruso, G. (2016). The Prescribed Curvature Problem in Low Dimension. In: Castrillón López, M., Hernández Encinas, L., Martínez Gadea, P., Rosado María, M. (eds) Geometry, Algebra and Applications: From Mechanics to Cryptography. Springer Proceedings in Mathematics & Statistics, vol 161. Springer, Cham. https://doi.org/10.1007/978-3-319-32085-4_4

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