Abstract
In this paper, we consider conformal characterizations of standard sphere in terms of conformal vector fields on closed Riemannian manifolds. We firstly prove that each closed Riemannian manifold with Ricci curvature being non-negative in certain direction and constant scalar curvature is isometric to standard sphere if and only if it admits a non-trivial closed conformal vector field. In the case of non-constant scalar curvature, we show that each closed Riemannian manifold of dimension two with positive Gauss curvature carrying a non-trivial closed conformal vector field is conformal to a round sphere and we generalize the result to high dimensions in two directions.
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References
Yano, K.: Integral formulas in Riemannian geometry. New York (1970)
Yano, K.: On harmonic and Killing vector fields. Annal. Math. 55, 38–45 (1952)
Yano, K., Nagano, T.: Einstein spaces admitting a one-parameter group of conformal transformations. Annl. Math. 69, 451–461 (1959)
Obata, M.: Certain conditions for a Riemannian manifold to be isometric with a sphere. J. Math. Soc. Japan 14, 333–340 (1962)
Bourguignon, J.P., Ezin, J.P.: Scalar curvature functions in a conformal class of metrics and conformal transformations. Trans. Am. Math. Soc. 301, 723–736 (1987)
Ejiri, N.: A negative answer to a conjecture of conformal transformations of Riemannian manifolds. J. Math. Soc. Japan 33, 261–266 (1981)
Bishop, R.L., Goldberg, S.I.: A characterization of the Euclidean sphere. Bull. Am. Math. Soc. 72, 122–124 (1966)
Nagano, T.: The conformal transformation on a space with parallel Ricci tensor. J. Math. Soc. Japan 11, 104 (1959)
Tanno, S., Weber, W.: Closed conformal vector fields. J. Diff. Geom. 3, 361–366 (1969)
Xu, X.: On the existence and uniqueness of solutions of Möbius equations. Trans. Am. Math. Soc. 337, 927–945 (1993)
Yano, K.: On Riemannian manifolds with constant scalar curvature admitting a conformal transformation group. Proc. Nat. Acad. Sci. USA 55, 472–476 (1966)
Deshmukh, S.: Characterizing spheres by conformal vector fields. Ann. Univ. Ferrara 56, 231–236 (2010)
Deshmukh, S., Turki, N.B.: A note on \(\varphi \) -analytic conformal vector fields. Anal. Math. Phys. 9, 181–195 (2019)
Deshmukh, S., Al-Solamy, F.: Characterizing spheres and Euclidean spaces by conformal vector field. Ann. Mat. Pura Appl. 196, 2135–2145 (2017)
Deshmukh, S.: Characterizations of Einstein manifolds and odd-dimensional spheres. J. Geom. Phys. 61(11), 2058–2063 (2011)
Deshmukh, S., Al-Solamy, F.: A note on conformal vector fields on a Riemannian manifold. Colloquium Mathematicum. 136(1), 65–73 (2014)
Deshmukh, S., Al-Solamy, F.: Conformal vector fields and conformal transformations on a Riemannian manifold. Balkan J. Geom. Appl. 17, 9–16 (2012)
Petersen, P.: Riemannian geometry, vol. 171. Springer, New York (2006)
Tashiro, Y.: Complete Riemannian manifolds and some vector fields. Trans. Am. Math. Soc. 117, 251–275 (1965)
Huang, L., Mo, X.: On conformal fields of a Randers metric with isotropic Scurvature. Illinois J. Math. 57(3), 685–696 (2013)
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The authors are grateful to the anonymous referee.
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Ye, J. Some characterizations of spheres by conformal vector fields. Ann Univ Ferrara 69, 49–58 (2023). https://doi.org/10.1007/s11565-022-00400-1
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DOI: https://doi.org/10.1007/s11565-022-00400-1