Abstract
Taking clue from the analytic vector fields on a complex manifold, \(\varphi \hbox {-analytic}\) conformal vector fields are defined on a Riemannian manifold (Deshmukh and Al-Solamy in Colloq. Math. 112(1):157–161, 2008). In this paper, we use \(\varphi \hbox {-analytic}\) conformal vector fields to find new characterizations of the n-sphere \( S^{n}(c)\) and the Euclidean space \((R^{n},\left\langle ,\right\rangle )\).
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References
Besse, A.L.: Einstein Manifolds. Springer, Berlin (1987)
Blair, D.E.: Contact Manifolds in Riemannian Geometry, Lecture Notes in Mathematics. Springer, Berlin (1976)
Blair, D.E.: On the characterization of complex projective space by differential equations. J. Math. Soc. Japan 27(1), 9–19 (1975)
Chavel, I.: Eigenvalues in Riemannian geometry. Academic press, Cambridge (1984)
Deshmukh, S.: Conformal vector fields and eigenvectors of Laplacian operator. Math. Phys. Anal. Geom. 15, 163–172 (2012)
Deshmukh, S.: Characterizing spheres by conformal vector fields. Ann. Univ. Ferrara 56(2), 231–236 (2010)
Deshmukh, S.: Conformal vector fields on Kaehler manifolds. Ann. Univ. Ferrara 57(1), 17–26 (2011)
Deshmukh, S., Al-Solamy, F.: Gradient conformal vector fields on a compact Riemannian manifold. Colloq. Math. 112(1), 157–161 (2008)
Deshmukh, S., Alsolamy, F.: A note on conformal vector fields on a Riemannian manifold. Colloq. Math. 136(1), 65–73 (2014)
Deshmukh, S., Alsolamy, F.: Conformal vector fields on a Riemannian manifold. Balk. J. Geom. Appl. 19(2), 86–93 (2014)
Deshmukh, S., Alsolamy, F.: Conformal vector fields and conformal transformations on a Riemannian manifold. Balk. J. Geom. Appl. 17(1), 9–16 (2012)
Deshmukh, S., Al-Eid, A.: Curvature bounds for the spectrum of a compact Riemannian manifold of constant scalar curvature. J. Geom. Anal. 15(4), 589–606 (2005)
Erkekoğlu, F., García-Río, E., Kupeli, D.N., Ünal, B.: Characterizing specific Riemannian manifolds by differential equations. Acta Appl. Math. 76(2), 195–219 (2003)
García-Río, E., Kupeli, D.N., Ünal, B.: Some conditions for Riemannian manifolds to be isometric with Euclidean spheres. J. Differ. Equ. 194(2), 287–299 (2003)
Obata, M.: Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Jpn. 14, 333–340 (1962)
Obata, M.: Riemannian manifolds admitting a solution of a certain system of differential equations. In: Proceedings of United States-Japan Seminar in Differential Geometry, Kyoto, pp. 101–114 (1965)
Obata, M.: Conformal transformations of Riemannian manifolds. J. Differ. Geom. 4, 311–333 (1970)
O’Neill, B.: Semi-Riemannian Geometry with Applications to Relativity. Academic Press, London (1983)
Pigola, S., Rimoldi, M., Setti, A.G.: Remarks on non-compact gradient Ricci solitons. Math. Z. 268(3–4), 777–790 (2011)
Tanno, S.: Some differential equations on Riemannian manifolds. J. Math. Soc. Jpn. 30(3), 509–531 (1978)
Tanno, S., Weber, W.: Closed conformal vector fields. J. Differ. Geom. 3, 361–366 (1969)
Tashiro, Y.: Complete Riemannian manifolds and some vector fields. Trans. Am. Math. Soc. 117, 251–275 (1965)
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We express our sincere gratitude to the referee for many helpful suggestions that lead to the improvement of this paper.
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This Work is supported by King Saud University, Deanship of Scientific Research, College of Science, Research Center.
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Deshmukh, S., Bin Turki, N. A note on \(\varphi \)-analytic conformal vector fields. Anal.Math.Phys. 9, 181–195 (2019). https://doi.org/10.1007/s13324-017-0190-8
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DOI: https://doi.org/10.1007/s13324-017-0190-8