Abstract
In this paper, we find a conformal vector field as well as a Killing vector field on a compact real submanifold of the canonical complex space form (Cm, J, h , i). In particular, using immersion ψ : M → Cm of a compact real submanifold M and the complex structure J of the canonical complex space form (Cm, J, h , i), we find conditions under which the tangential component of Jψ is a conformal vector field as well as conditions under which it is a Killing vector field. Finally, we obtain a characterization of n-spheres in the canonical complex space form (Cm, J, h , i).
Highlights
Conformal vector fields and Killing vector fields play a vital role in geometry of a Riemannian manifold (M, g) as well as in physics
Given an n-dimensional real submanifold (M, g) of the canonical complex space form (Cm, J, ) with immersion ψ : M → Cm, we treat ψ as the position vector field of points on M in Cm, and we have the expression Jψ = v + N, where v is the tangential component and N is the normal component of Jψ on M
We study the above mentioned question for real submanifolds of the canonical complex space form (Cm, J, ) and obtain conditions under which the vector field v is a conformal vector field (Theorems 3.1, 3.2) or a Killing vector field (Theorems 4.1, 4.3)
Summary
Conformal vector fields and Killing vector fields play a vital role in geometry of a Riemannian manifold (M, g) as well as in physics (cf. [13]). Given an n-dimensional real submanifold (M, g) of the canonical complex space form (Cm, J, , ) with immersion ψ : M → Cm, we treat ψ as the position vector field of points on M in Cm, and we have the expression Jψ = v + N , where v is the tangential component and N is the normal component of Jψ on M. This gives a globally defined vector field v on the real submanifold M. At the end of this paper, we give an example of a real submanifold of (Cm, J, , ) on which v is a nontrivial conformal vector field (that is, v is not Killing) and another example of a real submanifold on which v is nontrivial Killing vector field (that is, non-parallel)
Sign-in/Register to view highlights & summary Sign-in/Register to access full text options 
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Schedule a callDisclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.
