Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

RIEMANNIAN SUBMANIFOLDS WITH CONCIRCULAR CANONICAL FIELD

  • Received : 2018.12.19
  • Accepted : 2019.02.27
  • Published : 2019.11.30
Download PDF
⟨ Previous Next ⟩

Abstract

Let ${\tilde{M}}$ be a Riemannian manifold equipped with a concircular vector field ${\tilde{X}}$ and M a submanifold (with its induced metric) of ${\tilde{M}}$. Denote by X the restriction of ${\tilde{X}}$ on M and by $X^T$ the tangential component of X, called the canonical field of M. In this article we study submanifolds of ${\tilde{M}}$ whose canonical field $X^T$ is also concircular. Several characterizations and classification results in this respect are obtained.

Keywords

References

  1. R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc. 145 (1969), 1-49. https://doi.org/10.2307/1995057
  2. B.-Y. Chen, Pseudo-Riemannian Geometry, ${\delta}$-Invariants and Applications, World Scientic Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. https://doi.org/10.1142/9789814329644
  3. B.-Y. Chen, A simple characterization of generalized Robertson-Walker spacetimes, Gen. Relativity Gravitation 46 (2014), no. 12, Art. 1833, 5 pp. https://doi.org/10.1007/s10714-014-1833-9
  4. B.-Y. Chen, Some results on concircular vector elds and their applications to Ricci solitons, Bull. Korean Math. Soc. 52 (2015), no. 5, 1535-1547. https://doi.org/10.4134/BKMS.2015.52.5.1535
  5. B.-Y. Chen, Differential geometry of rectifying submanifolds, Int. Electron. J. Geom. 9 (2016), no. 2, 1-8; Addendum, ibid 10 (2017), no. 1, 81-82.
  6. B.-Y. Chen, Dierential Geometry of Warped Product Manifolds and Submanifolds, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017. https://doi.org/10.1142/10419
  7. B.-Y. Chen, Topics in differential geometry associated with position vector elds on Euclidean submanifolds, Arab J. Math. Sci. 23 (2017), no. 1, 1-17. https://doi.org/10.1016/j.ajmsc.2016.08.001
  8. B.-Y. Chen, Euclidean submanifolds via tangential components of their position vector elds, Mathematics 5 (2017), no. 4, Art. 51, pp. 17. https://doi.org/10.3390/math5040051
  9. B.-Y. Chen, Euclidean submanifolds with incompressible canonical vector field, Serdica Math. J. 43 (2017), no. 3-4, 321-334.
  10. B.-Y. Chen, A link between harmonicity of 2-distance functions and incompressibility of canonical vector fields, Tamkang J. Math. 49 (2018), no. 4, 339-347. https://doi.org/10.5556/j.tkjm.49.2018.2804
  11. B.-Y. Chen, Geometry of Submanifolds, Dover Publications, Mineola, NY, 2019.
  12. B.-Y. Chen and S. Deshmukh, Euclidean submanifolds with conformal canonical vector eld, Bull. Korean Math. Soc. 55 (2018), no. 6, 1823-1834. https://doi.org/10.4134/BKMS.b171100
  13. B.-Y. Chen and L. Verstraelen, A link between torse-forming vector fields and rotational hypersurfaces, Int. J. Geom. Methods Mod. Phys. 14 (2017), no. 12, 1750177, 10 pp. https://doi.org/10.1142/S0219887817501778
  14. B.-Y. Chen and S. W. Wei, Differential geometry of submanifolds of warped product manifolds I $\times$ $_f$ $S^-m-1}$(k), J. Geom. 91 (2009), no. 1-2, 21-42. https://doi.org/10.1007/s00022-008-2035-y
  15. B.-Y. Chen and S. W. Wei, Differential geometry of concircular submanifolds of Euclidean spaces, Serdica Math. J. 43 (2017), no. 1, 35-48.
  16. B.-Y. Chen and S. W. Wei, Sharp growth estimates for warping functions in multiply warped product manifolds, arXiv:1809.05737v1.
  17. A. Fialkow, Conformal geodesics, Trans. Amer. Math. Soc. 45 (1939), no. 3, 443-473. https://doi.org/10.2307/1990011
  18. Y. B. Han and S. W. Wei, $\Phi$-harmonic maps and $\Phi$-superstrongly unstable manifolds, preprint.
  19. R. Howard and S. W. Wei, Nonexistence of stable harmonic maps to and from certain homogeneous spaces and submanifolds of Euclidean space, Trans. Amer. Math. Soc. 294 (1986), no. 1, 319-331. https://doi.org/10.2307/2000133
  20. R. Howard and S. W. Wei, On the existence and nonexistence of stable submanifolds and currents in positively curved manifolds and the topology of submanifolds in Euclidean spaces, in Geometry and topology of submanifolds and currents, 127-167, Contemp. Math., 646, Amer. Math. Soc., Providence, RI, 2015. https://doi.org/10.1090/conm/646/12978
  21. K. Nomizu and K. Yano, On circles and spheres in Riemannian geometry, Math. Ann. 210 (1974), 163-170. https://doi.org/10.1007/BF01360038
  22. B. O'Neill, Semi-Riemannian Geometry with Applications to Relativity, Academic Press, New York, 1983.
  23. H. Takeno, Concircular scalar field in spherically symmetric space-times, I, Tensor (N.S.) 20 (1967) no. 2, 167-176.
  24. S. W. Wei, On topological vanishing theorems and the stability of Yang-Mills fields, Indiana Univ. Math. J. 33 (1984), no. 4, 511-529. https://doi.org/10.1512/iumj.1984.33.33027
  25. S. W. Wei, Representing homotopy groups and spaces of maps by p-harmonic maps, Indiana Univ. Math. J. 47 (1998), no. 2, 625-670. https://doi.org/10.1512/iumj.1998.47.1179
  26. S. W. Wei and C.-M. Yau, Regularity of p-energy minimizing maps and p-superstrongly unstable indices, J. Geom. Anal. 4 (1994), no. 2, 247-272. https://doi.org/10.1007/BF02921550
  27. K. Yano, Concircular geometry. I. Concircular transformations, Proc. Imp. Acad. Tokyo 16 (1940), 195-200. http://projecteuclid.org/euclid.pja/1195579139 https://doi.org/10.3792/pia/1195579139
  28. K. Yano, The Theory of Lie Derivatives and Its Applications, North-Holland Publishing Co., Amsterdam, 1957.