A link between harmonicity of 2-distance functions and incompressibility of canonical vector fields

Bang-Yen Chen

Abstract


Let $M$ be a Riemannian submanifold of a Riemannian manifold $\tilde M$ equipped with a concurrent vector field $\tilde Z$. Let $Z$ denote the restriction of $\tilde Z$ along $M$ and let $Z^T$ be the tangential component of $Z$ on $M$, called the canonical vector field of $M$. The 2-distance function $\delta^2_Z$ of $M$ (associated with $Z$) is defined by $\delta^2_Z=\$. In this article, we initiate the study of submanifolds $M$ of $\tilde M$ with incompressible canonical vector field $Z^T$ arisen from a concurrent vector field $\tilde Z$ on the ambient space $\tilde M$. First, we derive some necessary and sufficient conditions for such canonical vector fields to be incompressible. In particular, we prove that the 2-distance function $\delta^2_Z$ is harmonic if and only if the canonical vector field $Z^T$ on $M$ is an incompressible vector field. Then we provide some applications of our main results.

Keywords


Concurrent vector field; Canonical vector field; Incompressible vector field; Distance function; Harmonic function.

Full Text:

PDF

References


R. Aris, Vectors, Tensors, and the Basic Equations of Fluid Mechanics, Dover Publ., New York, 1989.

R. L. Bishop and B. O'Neill, Manifolds of negative curvature, Trans. Amer. Math. Soc., 145(1969), 1--49.

B.-Y. Chen, Pseudo-Riemannian Geometry, $delta$-Invariants and Applications, World Scientific Publishing, Hackensack, NJ, 2011.

B.-Y. Chen, Total Mean Curvature and Submanifolds of Finite Type, 2nd Edition, World Scientific Publishing, Hackensack, NJ, 2015.

B.-Y. Chen, Some results on concircular vector fields and their applications to Ricci solitons, Bull. Korean Math. Soc.,52(2015), 1535--1547.

B.-Y. Chen, Differential geometry of rectifying submanifolds, Int. Electron. J. Geom., 9(2016), 1--8.

B.-Y. Chen, Differential Geometry of Warped Product Manifolds and Submanifolds, World Scientific Publishing, Hackensack, NJ, 2017.

B.-Y. Chen, Addendum to: Differential geometry of rectifying submanifolds, Int. Electron. J. Geom., 10(2017), 81--82.

B.-Y. Chen, Topics in differential geometry associated with position vector fields on Euclidean submanifolds, Arab J. Math. Sci.23(2017), 1--17.

B.-Y. Chen, Euclidean submanifolds via tangential components of their position vector fields, Mathematics 5(2017), Art.51, pp.17.

B.-Y. Chen, Euclidean submanifolds with incompressible canonical vector field, Serdica Math. J., 43 (2017), 321--334.

B.-Y. Chen and S. Deshmukh, Euclidean submanifolds with conformal canonical vector field, Bull. Korean Math. Soc., 55 (2018) (in press).

B.-Y. Chen and L. Verstraelen, A link between torse-forming vector fields and rotational hypersurfaces, Int. J. Geom. Methods Mod. Phys., 14 (2017), Art. 1750177, pp.10.

B.-Y. Chen and S. W. Wei, Differential geometry of concircular submanifolds of Euclidean space,Serdica Math. J., 43 (2017), 45--58.

P. A. Davidson, Introduction to Magnetohydrodynamics, 2nd edition, Cambridge University Press, Cambridge, 2017.

J. Marsden and A. Tromba, Vector calculus, 5th Ed., W. H. Freedman and Company, New York, 2003.

M. Matsumoto and K. Eguchi, Finsler spaces admitting a concurrent vector field, Tensor (N.S.), 28 (1974), 239--249.

S.-I. Tachibana, On Finsler spaces which admit a concurrent vector field, Tensor(N.S.), (1950), 1--5.

K. Yano, Sur le parallelisme et la concourance dans l'espace de Riemann, {Proc. Imp. Acad. Tokyo, 19 (1943), 189--197.

K. Yano and B.-Y. Chen, On the concurrent vector fields of immersed manifolds, Kodai Math. Sem. Rep., 23(1971), 343--350.




DOI: http://dx.doi.org/10.5556/j.tkjm.49.2018.2804

Sponsored by Tamkang University | ISSN 0049-2930 (Print), ISSN 2073-9826 (Online) | Powered by MathJax