Abstract
Some characterizations of certain rank-one symmetric Riemannian manifolds by the existence of nontrivial solutions to certain partial differential equations on Riemannian manifolds are surveyed.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Alekseevsky, D. V. and Marchiafava, S.: Transformations of a quaternionic Kähler manifold, C. R. Acad. Sci. Paris, Sér. I 320 (1995), 703–708.
Alekseevsky, D. V. and Marchiafava, S.: Quaternionic transformations of a non-positive quaternionic Kähler manifold, Internat. J. Math. 8 (1997), 301–316.
Arikan, H. and Kupeli, D. N.: Some sufficient conditions for a map to be harmonic, Publ. Math. Debrecen 57(3–4) (2000), 489–498.
Blair, D. E.: On the characterization of complex projective space by differential equations, J. Math. Soc. Japan 27(1) (1975), 9–19.
Blair, D. E. and Showers, D. K.: A note on quaternionic geometry, Michigan Math. J. 23 (1976), 331–335.
Clifton, Y. H. and Maltz, R.: The k-nullity space of curvature operator, Michigan Math. J. 17 (1970), 85–89.
Ejiri, N.: A negative answer to a conjecture of conformal transformations of Riemannian manifolds, J. Math. Soc. Japan 33 (1981), 261–266.
Erkeko¢glu, F., Kupeli, D. N. and Ñnal, B.: Some results related to the Laplacian on vector fields, to appear.
Ferus, D.: Totally geodesic foliations, Math. Ann. 188 (1970), 313–316.
García-Río, E., Kupeli, D. N. and Ñnal, B.: Some conditions for Riemannian manifolds to be isometric with Euclidean spheres, J. Differential Equations, to appear.
Kanai, M.: On a differential equation characterizing a Riemannian structure of a manifold, Tokyo J. Math. 6(1) (1983), 143–151.
Kühnel, W.: Conformal transformations between Einstein spaces, In: R. Kulkarni and U. Pinkall (eds), Conformal Geometry, Aspects of Math. E12, Vieweg, Braunschweig, 1988, pp. 105–146.
Lichnerowicz, A.: Sur des transformations conformes d'une variété riemannienne compacte, C. R. Acad. Sci. Paris 259 (1964), 697–700.
LeBrun, C. and Ye, Y. G.: Fano manifolds, contact structures and quaternionic geometry, Internat. J. Math. 6 (1995), 419–437.
Maeda, Y.: On a characterization of quaternionic projective space by differential equations, Kodai Math. Sem. Rep. 27 (1976), 421–431.
Margerin, C.: A sharp lower bound for the first eigenvalue of a quaternion-Kähler manifold, Preprint, Ècole Polytechnique, 1094.1193.
Molzon, R. and Pinney Mortensen, K.: A Characterization of complex projective space up to biholomorphic isometry, J. Geom. Anal. 7(4) (1997), 611–621.
Obata, M.: Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Japan 14(3) (1962), 333–340.
Obata, M.: Riemannian manifolds admitting a solution of a certain system of differential equations, Proc. United States-Japan Seminar in Differential Geometry, Kyoto, 1965, pp. 101–114.
Poor, W. A.: Differential Geometric Structures, McGraw-Hill, New York, 1981.
Ranjan, A. and Santhanam, G.: A generalization of Obata's theorem, J. Geom. Anal. 7(3) (1997), 357–375.
Stoll, W.: The characterization of strictly parabolic manifolds, Ann. Sc. Nor. Sup. Pisa IV(VII) (1980), 87–154.
Tanno, S.: Some differential equations on Riemannian manifolds, J. Math. Soc. Japan 30(3) (1978), 509–531.
Tashiro, Y.: Complete Riemannian manifolds and some vector fields, Trans. Amer. Math. Soc. 117 (1965), 251–275.
Yano, K.: Integral Formulas in Riemannian Geometry, Marcel Dekker, New York, 1970.
Yano, K.: On geodesic vector fields in a compact orientable Riemannian space, Comment. Math. Helv. 35(1) (1961), 55–64.
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Erkekoğlu, F., García-Río, E., Kupeli, D.N. et al. Characterizing Specific Riemannian Manifolds by Differential Equations. Acta Applicandae Mathematicae 76, 195–219 (2003). https://doi.org/10.1023/A:1022987819448
Issue Date:
DOI: https://doi.org/10.1023/A:1022987819448