Skip to main content
SpringerLink
Log in

Conformal Riemannian Maps between Riemannian Manifolds, Their Harmonicity and Decomposition Theorems

  • Published:
Acta Applicandae Mathematicae Aims and scope Submit manuscript

Abstract

Riemannian maps were introduced by Fischer (Contemp. Math. 132:331–366, 1992) as a generalization isometric immersions and Riemannian submersions. He showed that such maps could be used to solve the generalized eikonal equation and to build a quantum model. On the other hand, horizontally conformal maps were defined by Fuglede (Ann. Inst. Fourier (Grenoble) 28:107–144, 1978) and Ishihara (J. Math. Kyoto Univ. 19:215–229, 1979) and these maps are useful for characterization of harmonic morphisms. Horizontally conformal maps (conformal maps) have their applications in medical imaging (brain imaging)and computer graphics. In this paper, as a generalization of Riemannian maps and horizontally conformal submersions, we introduce conformal Riemannian maps, present examples and characterizations. We show that an application of conformal Riemannian maps can be made in weakening the horizontal conformal version of Hermann’s theorem obtained by Okrut (Math. Notes 66(1):94–104, 1999). We also give a geometric characterization of harmonic conformal Riemannian maps and obtain decomposition theorems by using the existence of conformal Riemannian maps.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Abraham, R., Marsden, J.E., Ratiu, T.: Manifolds, Tensor Analysis and Applications. Springer, Berlin (1988)

    MATH  Google Scholar 

  2. Apostolova, L.G., Thompson, P.M.: Brain mapping as a tool to study neurodegeneration, neurotherapeutics. J. Am. Soc. Exp. Neurother. 4, 387–400 (2007)

    Google Scholar 

  3. Baird, P., Wood, J.C.: Harmonic Morphisms between Riemannian Manifolds. Clarendon Press, Oxford (2003)

    Book  MATH  Google Scholar 

  4. Beem, J.K., Ehrlich, P.E.: Global Lorentzian Geometry, 1st edn. Dekker, New York (1981). 2nd edn. (with Easley, K.L.) (1996)

    MATH  Google Scholar 

  5. Bishop, R.L., O’Neill, B.: Manifolds of negative curvature. Trans. Am. Math. Soc. 145, 1–49 (1969)

    Article  MATH  MathSciNet  Google Scholar 

  6. Chen, B.Y.: Totally umbilical submanifolds. Soochow J. Math. 5, 9–37 (1979)

    MathSciNet  Google Scholar 

  7. Chen, Q., Jost, J., Li, J., Wang, G.: Regularity theorems and energy identities for Dirac-harmonic maps. Math. Z. 251(1), 61–84 (2005)

    Article  MATH  MathSciNet  Google Scholar 

  8. Chen, Q., Jost, J., Li, J., Wang, G.: Dirac-harmonic maps. Math. Z. 254(2), 409–432 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  9. De Rham, G.: Sur la réducibilité dùn espace de Riemann. Comment. Math. Helv. 26, 328–344 (1952)

    Article  MATH  MathSciNet  Google Scholar 

  10. Dubrovin, B.A., Fomenko, A.T., Novikov, S.P.: Modern Geometry, Methods and Applications. Part II. The Geometry and Topology of Manifolds. Graduate Texts in Mathematics, vol. 104. Springer, Berlin (1985)

    Google Scholar 

  11. Eells, J., Sampson, H.J.: Harmonic mappings of Riemannian manifolds. Am. J. Math. 86, 109–160 (1964)

    Article  MATH  MathSciNet  Google Scholar 

  12. Falcitelli, M., Ianus, S., Pastore, A.M.: Riemannian Submersions and Related Topics. World Scientific, Singapore (2004)

    MATH  Google Scholar 

  13. Fischer, A.E.: Riemannian maps between Riemannian manifolds. Contemp. Math. 132, 331–366 (1992)

    Google Scholar 

  14. Fischer, A.E.: Riemannian submersions and the regular interval theorem of Morse theory. Ann. Glob. Anal. Geom. 14(3), 263–300 (1996)

    Article  MATH  Google Scholar 

  15. Fuglede, B.: Harmonic morphisms between Riemannian manifolds. Ann. Inst. Fourier (Grenoble) 28, 107–144 (1978)

    MATH  MathSciNet  Google Scholar 

  16. Garcia-Rio, E., Kupeli, D.N.: Singularity versus splitting theorems for stably causal spacetimes. Ann. Glob. Anal. Geom. 14(3), 301–312 (1996)

    Article  MATH  MathSciNet  Google Scholar 

  17. Garcia-Rio, E., Kupeli, D.N.: Semi-Riemannian Maps and Their Applications. Kluwer Academic, Dordrecht (1999)

    MATH  Google Scholar 

  18. Gray, A.: Pseudo-Riemannian almost product manifolds and submersions. J. Math. Mech. 16, 715–737 (1967)

    MATH  MathSciNet  Google Scholar 

  19. Gu, X., Wang, Y., Yau, S.T.: Computing conformal invariants:periodic matrices. Commun. Inf. Syst. 3(3), 153–169 (2003)

    MATH  MathSciNet  Google Scholar 

  20. Gu, X., Wang, Y., Chan, T.F., Thompson, P.M., Yau, S.T.: Genus zero surface conformal mapping and its applications to brain surface mapping. IEEE Trans. Med. Imaging 23(8), 949–958 (2004)

    Article  Google Scholar 

  21. Hawking, S., Ellis, G.F.R.: The Large Scale Structure of Spacetime. Cambridge University Press, Cambridge (1973)

    Google Scholar 

  22. Hermann, R.: A sufficient condition that a mapping of Riemannian manifolds be a fibre bundle. Proc. Am. Math. Soc. 11, 236–242 (1960)

    Article  MATH  MathSciNet  Google Scholar 

  23. Hiepko, S.: Eine innere Kennzeichnung der verzerrten produkte. Math. Ann. 241, 209–215 (1979)

    Article  MATH  MathSciNet  Google Scholar 

  24. Hurdal, M.K., Kurtz, K.W., Banks, D.C.: Case study: interacting with cortical flat maps of the human brain. In: Proceedings Visualization 2001, pp. 469–472. IEEE, Piscataway (2001)

    Chapter  Google Scholar 

  25. Ishihara, T.: A mapping of Riemannian manifolds which preserves harmonic functions. J. Math. Kyoto Univ. 19, 215–229 (1979)

    MATH  MathSciNet  Google Scholar 

  26. Kock, A.: A geometric theory of harmonic and semi-conformal maps. Cent. Eur. J. Math. 2(5), 708–724 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  27. Miao, J., Wang, Y., Gu, X., Yau, S.T.: Optimal global conformal surface parametrization for visualization. Commun. Inf. Syst. 4(2), 117–134 (2005)

    MathSciNet  Google Scholar 

  28. Mustafa, M.T.: Applications of harmonic morphisms to gravity. J. Math. Phys. 41(10), 6918–6929 (2000)

    Article  MATH  MathSciNet  Google Scholar 

  29. Nore, T.: Second fundamental form of a map. Ann. Mat. Pura Appl. 4(146), 281–310 (1987)

    MathSciNet  Google Scholar 

  30. Okrut, S.I.: Generalized Hermann theorems and conformal holomorphic submersions. Math. Notes 66(1), 94–104 (1999)

    Article  MATH  MathSciNet  Google Scholar 

  31. O’Neill, B.: The fundamental equations of a submersion. Mich. Math. J. 13, 459–469 (1966)

    Article  MATH  MathSciNet  Google Scholar 

  32. Ponge, R., Reckziegel, H.: Twisted products in pseudo Riemannian geometry. Geom. Dedic. 48, 15–25 (1993)

    Article  MATH  MathSciNet  Google Scholar 

  33. Sachs, R.K., Wu, H.: General Relativity for Mathematicians. Springer, Berlin (1977)

    MATH  Google Scholar 

  34. Schwinger, J.: A theory of the fundamental interactions. Ann. Phys. 2, 407–434 (1957)

    Article  MATH  MathSciNet  Google Scholar 

  35. Stepanov, S.E.: On the global theory of some classes of mapping. Ann. Glob. Anal. Geom. 13, 239–249 (1995)

    Article  MATH  Google Scholar 

  36. Wang, Y., Gu, X., Yau, S.T.: Volumetric harmonic map. Commun. Inf. Syst. 3(3), 191–201 (2003)

    MATH  MathSciNet  Google Scholar 

  37. Wang, Y., Lui, L.M., Chan, T.F., Thompson, P.M.: In: Optimization of Brain Conformal Mapping Using Landmarks, Medical Image Computing and Computer Assisted Interventions (MICCAI), Part II, pp. 675–683, Palm Springs, CA, 26–29 Oct. 2005

  38. Wang, Y., Gu, X., Chan, T.F., Thompson, P.M., Yau, S.T.: Brain surface conformal parametrization with the Ricci flow. In: IEEE International Symposium on Biomedical Imaging—From Nano to Macro (ISBI), pp. 1312–1315, Washington D.C. (2007)

  39. Wang, Y., Yin, X., Zhang, J., Gu, X., Chan, T.F., Thompson, P.M., Yau, S.T.: In: Brain Mapping with the Ricci Flow Conformal Parameterization and Multivariate Statistics on Deformation Tensors, 2nd MICCAI Workshop on Mathematical Foundations of Computational Anatomy, New York, pp. 36–47 (2008)

  40. Watson, B.: Riemannian submersions and instantons. Frontiers of Applied Geometry, Las Cruces, N.M., 1980. Math. Model. 1 (1980), no. 4 (1981), pp. 381–393

  41. Yano, K., Kon, M.: Structures on Manifolds. Ser. Pure Math. World Scientific, Singapore (1984)

    Google Scholar 

  42. Zhang, D., Hebert, M.: Harmonic maps and their applications in surface matching. In: Proc. CVPR’99 (Computer Vision and Pattern Recognition), Colorado, June 1999

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Inonu University, 44280, Malatya, Turkey

    Bayram Ṣahin

Authors
  1. Bayram Ṣahin
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bayram Ṣahin.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Ṣahin, B. Conformal Riemannian Maps between Riemannian Manifolds, Their Harmonicity and Decomposition Theorems. Acta Appl Math 109, 829–847 (2010). https://doi.org/10.1007/s10440-008-9348-6

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10440-008-9348-6

Keywords

Mathematics Subject Classification (2000)

Search

Navigation