Abstract
In this article there are found precise upper bounds of dimension of vector spaces of conformal Killing forms, closed and coclosed conformal Killing \(r\)-forms \((1\,{\le }\, r\,{\le }\, n {-} 1)\) on an \(n\)-dimensional manifold. It is proved that, in the case of \(n\)-dimensional closed Riemannian manifold, the vector space of conformal Killing \(r\)-forms is an orthogonal sum of the subspace of Killing forms and of the subspace of exact conformal Killing \(r\)-forms. This is a generalization of related local result of Tachibana and Kashiwada on pointwise decomposition of conformal Killing \(r\)-forms on a Riemannian manifold with constant curvature. It is shown that the following well known proposition may be derived as a consequence of our result: any closed Riemannian manifold having zero Betti number and admitting group of conformal mappings, which is non equal to the group of motions, is conformal equivalent to a hypersphere of Euclidean space.
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References
Alodan, H.: Conformal gradient vector fields. Differential Geometry—Dynamical System 12, 1–3 (2010)
Besse, A.L.: Einstein Manifolds. Springer, Berlin (1987)
Besse, A.L.: Géométrie Riemannienne en dimension 4. Cedic-Fernand Nathan, Paris (1981)
Benn, I.M., Chalton, P.: Dirac symmetry operators from conformal Killing-Yano tensors. Class. Quantum Gravity 14(5), 1037–1042 (1997)
Eisenhart, L.P.: Riemannian Geometry. Princeton University Press, Princeton (1926)
Kühnel, W., Rademacher, H.-B.: Conformal transformations of pseudo-Riemannian manifolds. Recent developments in pseudo-Riemannian geometry. Zürich Eur. Math. Soc. 261–298 (2008)
Kashiwada, T.: On conformal Killing tensor. Nat. Sci. Rep. Ochanomizu Univ 19(2), 67–74 (1968)
Kora, M.: On conformal Killing forms and the proper space of for p-forms. Math. J. Okayama Univ. 22, 195–204 (1980)
Lichnerowicz, A.: Géométrie des groupes de transformations. Dunod, Paris (1958)
Maccallum, M., Herlt, E., Schmutzer, E.E.: Exact Solutions of the Einstein Field Equations. Deutscher Verlag der Wissenschaften, Berlin (1980)
Mikeš, J., Vanžurová, A., Hinterleitner, I.: Geodesic Mappings and Some Generalizations. Palacký University Press, Olomouc (2009)
Novikov, S.P.: Topology I. Encycl. Math. Sci. 12, 1–310 (1996) (translation from Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 12, 5–252 (1986))
Petersen, P.: Riemannian Geometry. Springer, New York (1997)
Pierzchalski, A.: Ricci curvature and quasiconformal deformations of a Riemannian manifold. Manuscripta Math. 66, 113–127 (1989)
De Rham, G.: Variétés Différentiables. Hermann, Paris (1955)
Stepanov, S.E.: A new strong Laplacian on differential forms. Math. Notes 76(3), 420–425 (2004)
Stepanov, S.E.: Curvature and Tachibana numbers. Sbornik Math. 202(7), 135–146 (2011)
Stepanov, S.E.: The Killing-Yano tensor. Theoret. Math. Phys. 134(3), 333–338 (2003)
Stepanov, S.E.: The vector space of conformal Killing forms on a Riemannian manifold. J. Math. Sci. 110(4), 2892–2906 (2002)
Stepanov, S.E.: On conformal Killing 2-form of the electromagnetic field. J. Geom. Phys. 33, 191–209 (2000)
Stepanov, S.E.: Vanishing theorems in affine, Riemannian and Lorenz geometries. J. Math. Sci. 141(1), 929–964 (2007)
Semmelmann, U.: Conformal Killing forms on Riemannian manifolds. Math. Z. 245(3), 503–627 (2003)
Suyama, Y., Tsukamoto, Y.: Riemannian manifolds admitting a certain conformal transformation group. J. Diff. Geom. 5, 415–426 (1971)
Stepanov, S.E.: Some conformal and projective scalar invariants of Riemannian manifolds. Math. Notes 80(6), 848–852 (2006)
Tachibana, S.: On the proper space of for m-forms im 2m dimensional conformal flat Riemannian manifolds. Nat. Sci. Rep. Ochanomizu Univ. 28, 111–115 (1978)
Tachibana, S.: On conformal Killing tensor in a Riemannian space. Tohoku Math. J. 21, 56–64 (1969)
Tashiro, Y.: Complete Riemannian manifolds and some vector fields. Trans. Am. Math. Soc. 117, 251–275 (1965)
Tanno, S., Weber, W.C.: Closed conformal vector fields. J. Diff. Geom. 3, 361–366 (1969)
Wolf, J.A.: Space of Constant Curvature. California University Press, Berkley (1972)
Yano, K.: Concircular geometry. Proc. Imp. Acad. Tokyo 16, 195–200, 354–360, 442–448, 505–511 (1940)
Yano, K., Bochner, C.: Curvature and Betti Numbers. Princeton University Press, Princeton (1953)
Yano, K., Sawaki, S.: Riemannian manifolds admitting a conformal transformation group. J. Diff. Geom. 2, 161–184 (1968)
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The paper was supported by grant P201/11/0356 of The Czech Science Foundation.
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Stepanov, S.E., Jukl, M., Mikeš, J. (2014). On Dimensions of Vector Spaces of Conformal Killing Forms. In: Makhlouf, A., Paal, E., Silvestrov, S., Stolin, A. (eds) Algebra, Geometry and Mathematical Physics. Springer Proceedings in Mathematics & Statistics, vol 85. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-55361-5_29
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DOI: https://doi.org/10.1007/978-3-642-55361-5_29
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