On Dimensions of Vector Spaces of Conformal Killing Forms

Степанов, С. Е.; Jukl, Marek; Mikeš, Josef

doi:10.1007/978-3-642-55361-5_29

Cited by 2 publications

(4 citation statements)

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“…Therefore, the space of all co-closed conformal Killing r-forms we can define as [23]). Sometimes, closed conformal Killing forms are also called planar forms (see [16]). Therefore, the space of all closed conformal Killing r-forms we can define as P…”

Section: Preliminariesmentioning

confidence: 99%

“…(see [15] and [16]). The condition 2 1 ker ker D D    characterizes the form  as a harmonic form (see [19]; [24, pp.…”

Section: Preliminariesmentioning

confidence: 99%

“…An important invariant of a symmetric space is its rank which is the maximal dimension of a totally geodesic flat subspace. In particular, rank-one symmetric spaces are an important class among symmetric spaces (see [16] and [17]). In his book [17], Chavel gave a beautiful account of the rank-one symmetric spaces from a geometric point of view up to the classification of them, which he left for the reader to pursue as a matter in Lie group theory.…”

Section: Conformal Killing Forms On Riemannian Globally Symmetric Spacesmentioning

confidence: 99%

“…We prove the following theorem for conformal Killing forms on real rank-one symmetric spaces using his results. , respectively (see [15] and [16]). This proves our Theorem 3.…”

Section: Conformal Killing Forms On Riemannian Globally Symmetric Spacesmentioning

confidence: 99%

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Exterior Differential Forms on Riemannian Symmetric Spaces

Alexandrova¹,

Степанов²,

Tsyganok³

2017

Science Evolution

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In the present paper we give a rough classification of exterior differential forms on a Riemannian manifold. We define conformal Killing, closed conformal Killing, coclosed conformal Killing and harmonic forms due to this classification and consider these forms on a Riemannian globally symmetric space and, in particular, on a rank-one Riemannian symmetric space. We prove vanishing theorems for conformal Killing L 2-forms on a Riemannian globally symmetric space of noncompact type. Namely, we prove that every closed or co-closed conformal Killing L 2-form is a parallel form on an arbitrary such manifold. If the volume of it is infinite, then every closed or co-closed conformal Killing L 2-form is identically zero. In addition, we prove vanishing theorems for harmonic forms on some Riemannian globally symmetric spaces of compact type. Namely, we prove that all harmonic one-formsvanish everywhere and every harmonic r -form  r  2 is parallel on an arbitrary such manifold. Our proofs are based on the Bochnertechnique and its generalized version that are most elegant and important analytical methods in differential geometry “in the large”.

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Section: Preliminariesmentioning

confidence: 99%

“…(see [15] and [16]). The condition 2 1 ker ker D D    characterizes the form  as a harmonic form (see [19]; [24, pp.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Conformal Killing Forms On Riemannian Globally Symmetric Spacesmentioning

confidence: 99%

“…We prove the following theorem for conformal Killing forms on real rank-one symmetric spaces using his results. , respectively (see [15] and [16]). This proves our Theorem 3.…”

Section: Conformal Killing Forms On Riemannian Globally Symmetric Spacesmentioning

confidence: 99%

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