Theorems of existence and non-existence of conformal Killing forms

S. E. Stepanov; I. I. Tsyganok

Geodesic

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Theorems of existence and non-existence of conformal Killing forms

S. E. Stepanov ; I. I. Tsyganok

Izvestiâ vysših učebnyh zavedenij. Matematika, no. 10 (2014), pp. 54-61

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Résumé

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.

@article{IVM_2014_10_a6,
     author = {S. E. Stepanov and I. I. Tsyganok},
     title = {Theorems of existence and non-existence of conformal {Killing} forms},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {54--61},
     publisher = {mathdoc},
     number = {10},
     year = {2014},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IVM_2014_10_a6/}
}

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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/