Abstract
In this paper, we consider an n-dimensional compact Riemannian manifold (M,g) of constant scalar curvature and show that the presence of a non-Killing conformal vector field ξ on M that is also an eigenvector of the Laplacian operator acting on smooth vector fields with eigenvalue λ together with a condition on Ricci curvature of M, that the Ricci curvature in the direction of a certain vector field is greater than or equal to (n − 1)λ, forces M to be isometric to the n-sphere S n(λ).
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References
Amur, K., Hedge, V.S.: Conformality of Riemannian manifolds to spheres. J. Diff. Geom. 9, 571–576 (1974)
Blair, D.E.: Riemannian geometry of contact and symplectic manifolds. Progress in Mathematics 203, Birkhauser (2010)
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 (1984)
Deshmukh, S.: Characterizing spheres by conformal vector fields. Ann. Univ. Ferrara Sez. VII Sci. Mat. 56(2), 231–236 (2010)
Deshmukh, S., Al-Solamy, F.: Conformal gradient vector fields on a compact Riemannian manifold. Colloquium Math. 112(1), 157–161 (2008)
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)
Kanai, M.: On a differential equation characterizing a Riemannian structure of a manifold. Tokyo J. Math. 6(1), 143–151 (1983)
Lichnerowicz, A.: Sur des transformations conformes d’une variété riemannienne compacte. C.R. Acad. Sci. Paris 259, 697–700 (1964)
Molzon, R., Pinney Mortensen, K.: A characterization of complex projective space up to biholomorphic isometry. J. Geom. Anal. 7(4), 611–621 (1997)
Obata, M.: Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Japan 14, 333–340 (1962)
Obata, M.: Riemannian manifolds admitting a solution of a certain system of differential equations. In: Proc. United States–Japan Seminar in Differential Geometry, Kyoto, pp. 101–114 (1965)
Obata, M.: Conformal transformations of Riemannian manifolds, J. Diff. Geom. 4, 311–333 (1970)
Ranjan, A., Santhanam, G.: A generalization of Obata’s theorem. J. Geom. Anal. 7(3), 357–375 (1997)
Tanno, S.: Some differential equations on Riemannian manifolds. J. Math. Soc. Japan 30(3), 509–531 (1978)
Tanno, S., Weber, W.: Closed conformal vector fields. J. Diff. Geom. 3, 361–366 (1969)
Tashiro, Y.: Complete Riemannian manifolds and some vector fields. Trans. Amer. Math. Soc. 117, 251–275 (1965)
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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. Conformal Vector Fields and Eigenvectors of Laplacian Operator. Math Phys Anal Geom 15, 163–172 (2012). https://doi.org/10.1007/s11040-012-9106-x
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DOI: https://doi.org/10.1007/s11040-012-9106-x