• Received : 2017.12.20
  • Accepted : 2018.04.13
  • Published : 2018.11.30


The position vector field x is the most elementary and natural geometric object on a Euclidean submanifold M. The position vector field plays very important roles in mathematics as well as in physics. Similarly, the tangential component $x^T$ of the position vector field is the most natural vector field tangent to the Euclidean submanifold M. We simply call the vector field $x^T$ the canonical vector field of the Euclidean submanifold M. In earlier articles [4,5,9,11,12], we investigated Euclidean submanifolds whose canonical vector fields are concurrent, concircular, torse-forming, conservative or incompressible. In this article we study Euclidean submanifolds with conformal canonical vector field. In particular, we characterize such submanifolds. Several applications are also given. In the last section we present three global results on complete Euclidean submanifolds with conformal canonical vector field.


Supported by : King Saud University


  1. H. Alohali, H. Alodan, and S. Deshmukh, Conformal vector fields on submanifolds of a Euclidean space, Publ. Math. Debrecen 91 (2017), no. 1-2, 217-233.
  2. B.-Y. Chen, Geometry of Submanifolds, Marcel Dekker, Inc., New York, 1973.
  3. B.-Y. Chen, Pseudo-Riemannian Geometry, ${\delta}$-Invariants and Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011.
  4. B.-Y. Chen, Differential geometry of rectifying submanifolds, Int. Electron. J. Geom. 9 (2016), no. 2, 1-8.
  5. B.-Y. Chen, Addendum to: Differential geometry of rectifying submanifolds, Int. Electron. J. Geom. 10 (2017), no. 1, 81-82.
  6. B.-Y. Chen, Differential Geometry of Warped Product Manifolds and Submanifolds, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2017.
  7. B.-Y. Chen, Topics in differential geometry associated with position vector fields on Euclidean submanifolds, Arab J. Math. Sci. 23 (2017), no. 1, 1-17.
  8. B.-Y. Chen, Euclidean submanifolds via tangential components of their position vector fields, Mathematics 5 (2017), no. 4, Art. 51, pp. 17.
  9. B.-Y. Chen, Euclidean submanifolds with incompressible canonical vector field, Serdica Math. J. 43 (2017), 257-270.
  10. B.-Y. Chen and S. Deshmukh, Yamabe and quasi-Yamabe solitons on Euclidean submanifolds, arXiv:1711.02978v1 [math.DG] (2017).
  11. 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.
  12. B.-Y. Chen and S. W. Wei, Differential geometry of concircular submanifolds of Euclidean spaces, Serdica Math. J. 43 (2017), no. 1, 35-48.
  13. B.-Y. Chen and K. Yano, Integral formulas for submanifolds and their applications, J. Differential Geometry 5 (1971), 467-477.
  14. B.-Y. Chen and K. Yano, Umbilical submanifolds with respect to a nonparallel normal direction, J. Differential Geometry 8 (1973), 589-597.
  15. S. Deshmukh and I. Al-Dayel, Characterizing spheres by an immersion in Euclidean spaces, Arab J. Math. Sci. 23 (2017), no. 1, 85-93.
  16. R. S. Hamilton, The Ricci flow on surfaces, in Mathematics and general relativity (Santa Cruz, CA, 1986), 237-262, Contemp. Math., 71, Amer. Math. Soc., Providence, RI, 1988.
  17. M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14 (1962), 333-340.
  18. S. Pigola, M. Rimoldi, and A. G. Setti, Remarks on non-compact gradient Ricci solitons, Math. Z. 268 (2011), no. 3-4, 777-790.
  19. J. A. Schouten, Ricci-Calculus, 2nd ed., Berlin, Germany, Springer-Verlag, 1954.
  20. K. Yano, Concircular geometry. I. Concircular transformations, Proc. Imp. Acad. Tokyo 16 (1940), 195-200.
  21. K. Yano, On the torse-forming directions in Riemannian spaces, Proc. Imp. Acad. Tokyo 20 (1944), 340-345.
  22. K. Yano, The Theory of Lie Derivatives and Its Applications, North-Holland Publishing Co., Amsterdam, 1957.