Ivancevic_Applied-Diff-Geom

Applied Manifold Geometry 443⌋ is the operation dual to the wedge product on M. It is easy to see thata conformal Killing 1−form is dual to a conformal vector–field. Coclosedconformal Killing p−forms are called Killing forms. For p = 1 they aredual to Killing vector–fields.Let α be a Killing p−form and let γ be a geodesic on (M, g), i.e.,∇ ˙γ ˙γ = 0. Then∇ ˙γ ( ˙γ⌋ α) = (∇ ˙γ ˙γ)⌋ α + ˙γ⌋ ∇ ˙γ α = 0,i.e., ˙γ⌋ α is a (p − 1)−form parallel along the geodesic γ and in particularits length is constant along γ.The l.h.s of equation (3.245) defines a first–order elliptic differentialoperator T , the so–caled twistor operator. Equivalently one can describe aconformal Killing form as a form in the kernel of twistor operator T . Fromthis point of view conformal Killing forms are similar to Penrose’s twistorspinors in Lorentzian spin geometry. One shared property is the conformalinvariance of the defining equation. In particular, any form which is parallelfor some metric g, and thus a Killing form for trivial reasons, induces non–parallel conformal Killing forms for metrics conformally equivalent to g (bya non–trivial change of the metric) [Semmelmann (2002)].3.15.2 Conformal Killing Tensors and LaplacianSymmetryIn an nD Riemannian manifold (M, g), a Killing tensor–field (of order 2)is a symmetric tensor K ab satisfying (generalizing (3.244))K (ab;c) = 0. (3.246)A conformal Killing tensor–field (of order 2) is a symmetric tensor Q absatisfyingQ (ab;c) = q (a g bc) , with q a = (Q ,a + 2Q a;dd)/(n + 2), (3.247)where comma denotes partial derivative and Q = Q d d. When the associatedconformal vector q a is nonzero, the conformal Killing tensor will be calledproper and otherwise it is a (ordinary) Killing tensor. If q a is a Killingvector, Q ab is referred to as a homothetic Killing tensor. If the associatedconformal vector q a = q ,a is the gradient of some scalar field q, then Q abis called a gradient conformal Killing tensor. For each gradient conformal