Geodesic
Theorems of existence and non-existence of conformal Killing forms
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2014), pp. 54-61.

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On an $n$-dimensional compact, orientable, connected Riemannian manifold, we consider the curvature operator acting on the space of covariant traceless symmetric $2$-tensors. We prove that, if the curvature operator is negative, the manifold admits no nonzero conformal Killing $p$-forms for $p=1,2,\dots,n-1$. On the other hand, we prove that the dimension of the vector space of conformal Killing $p$-forms on an $n$-dimensional compact simply-connected conformally flat Riemannian manifold $(M, g)$ is not zero.
Keywords: Riemannian manifold, curvature operator, conformal Killing forms, vanishing theorem, existence theorem. 
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S. E. Stepanov; I. I. Tsyganok. Theorems of existence and non-existence of conformal Killing forms. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2014), pp. 54-61. http://geodesic.mathdoc.fr/item/IVM_2014_10_a6/

[1] Tachibana Sh., “On conformal Killing tensor in a Riemannian space”, Tôhoku Math. J., 21 (1969), 56–64 | DOI | MR | Zbl

[2] Kashiwada T., “On conformal Killing tensor”, Nat. Sci. Rep. Ochanomizu Univ., 19:2 (1968), 67–74 | MR | Zbl

[3] Kobayasi Sh., Nomidzu K., Osnovy differentsialnoi geometrii, v. 1, Nauka, M., 1981

[4] Yano K., Bokhner S., Krivizna i chisla Betti, In. lit., M., 1957

[5] Stepanov S. E., “Krivizna i chisla Tachibany”, Matem. sb., 202:7 (2011), 135–146 | DOI | MR | Zbl

[6] Novikov S. P., “Topologiya”, Itogi nauki tekhn. Sovremen. probl. matem. Fundamentalnye napravleniya, 12, VINITI AN SSSR, 1986, 5–252 | MR | Zbl

[7] Stepanov S. E., “O nekotorykh konformnykh i proektivnykh skalyarnykh invariantakh rimanova mnogoobraziya”, Matem. zametki, 80:6 (2006), 902–907 | DOI | MR | Zbl

[8] Kobayasi Sh., Nomidzu K., Osnovy differentsialnoi geometrii, v. 2, Nauka, M., 1981

[9] Petrsen P., Riemannian geometry, Springer Science, New York, 2006 | MR

[10] Besse A., Mnogoobraziya Einshteina, Nauka, M., 1990

[11] Nishikawa S., “On deformation of Riemannian metrics and manifolds with positive curvature operator”, Lecture Notes in Math., 1201, 1986, 202–211 | DOI | MR | Zbl

[12] Tachibana Sh., Ogiue K., “Les variétés riemanniennes dont l'opérateur de coubure restreint est positif sont des sphéres d'homologie réelle”, C. R. Acad. Sci. Paris, 289 (1979), 29–30 | MR | Zbl

[13] Semmelmann U., “Conformal Killing forms on Riemannian manifolds”, Math. Z., 245:3 (2003), 503–527 | DOI | MR | Zbl

[14] Kashiwada T., “On the curvature operator of the second kind”, Natural Science Report, Ochanomizu University, 44:2 (1993), 69–73 | MR | Zbl

[15] Kora M., “On conformal Killing forms and the proper space $\Delta$ for $p$-forms”, Math. J. Okayama Univ., 22:2 (1980), 195–204 | MR | Zbl

[16] Yano K., Integral formulas in Riemannian geometry, Marcel Dekker, New York, 1970 | MR | Zbl

[17] Stepanov S. E., “On conformal Killing $2$-form of the electromagnetic field”, J. Geom. and Phys., 33 (2000), 191–209 | DOI | MR | Zbl

[18] Tachibana Sh., Kashiwada T., “On the integrability of Killing–Yano's equation”, J. Math. Soc. Japan, 21:2 (1969), 259–265 | DOI | MR | Zbl