Killing forms on symmetric spaces

Abstract

Killing forms on Riemannian manifolds are differential forms whose covariant derivative is totally skew-symmetric. We show that a compact simply connected symmetric space carries a non-parallel Killing p-form (p ≥ 2) if and only if it isometric to a Riemannian product S k × N , where S k is a round sphere and k > p.2000 Mathematics Subject Classification: Primary 53C55, 58J50.

Conformally related Riemannian metrics with non-generic holonomy

“…In particular, if ψ is a closed conformal Killing form, * ψ is a Killing form. Killing forms on compact manifolds with reduced holonomy have been recently studied in a series of papers [2], [11], [12], [13], [17] and [18]. The following proposition summarizes some of the results of these papers.…”

“…1.1], where one further assumes that (M, g) is a simply connected symmetric space of compact type. This assumption is in fact superfluous in the irreducible case, since[2, Lemma 4.3] actually shows that if an irreducible locally symmetric space (M, g) carries a non-parallel Killing p-form with p ≥ 2, then its Weyl tensor vanishes identically, thus (M, g) has constant sectional curvature (being Einstein).…”

Invariant solutions to the conformal Killing–Yano equation on Lie groups

“…In [10] the authors prove that a compact simply connected symmetric space carries a non-parallel Killing-Yano p-form if and only if it is isometric to a Riemannian product S k × N, where S k is a round sphere and k > p. The case of conformal Killing-Yano p-forms on a compact Riemannian product was considered in [16], proving that such a form is a sum of forms of the following types: parallel forms, pull-back of Killing forms on the factors, and their Hodge duals. A family of examples of manifolds carrying CKY 2-forms is given by spheres, nearly Kähler manifolds and Sasakian manifolds.…”

“…More recently, since the work of Moroianu, Semmelmann [8,9], a renewed interest in the subject arose among differential geometers (see, for instance, [10][11][12][13][14][15][16]). …”

Nonlinear symmetries on spaces admitting Killing tensors

“…The covariantly constant K-Y tensors represent a particular class of K-Y tensors and they play a special role in the theory of Dirac operators as it will be seen in Section 4. We mention that K-Y tensors are also called Yano tensors or Killing forms, and CKY tensors are sometimes referred as conformal Yano tensors, conformal Killing forms or twistor forms [20][21][22].…”