Ivancevic_Applied-Diff-Geom

442 Applied Differential Geometry: A Modern IntroductionNow, to properly initialize our conformal geometry, recall that conformaltwistor spinor–fields ϕ were introduced by R. Penrose into physics (see[Penrose (1967); Penrose and Rindler (1984); Penrose and Rindler (1986)])as solutions of the conformally covariant twistor equation∇ S Xϕ + 1 X · Dϕ = 0,nfor each vector–fields X on a Riemannian manifold (M, g), where D is theDirac operator. Each twistor spinor–field ϕ on (M, g) defines a conformalvector–field V ϕ on M byg(V ϕ , X) = i k+1 〈X · ϕ, ϕ〉.Also, each twistor spinor–field ϕ that satisfies the Dirac equation on (M, g),Dϕ = µϕ,is called a Killing spinor–field. Each twistor spinor–field without zeros on(M, g) can be transformed by a conformal change of the metric g into aKilling spinor–field [Baum (2000)].3.15.1 Conformal Killing Vector–Fields and Forms on MThe space of all conformal Killing vector–fields forms the Lie algebra of theisometry group of a Riemannian manifold (M, g) and the number of linearlyindependent Killing vector–fields measures the degree of symmetry ofM. It is known that this number is bounded from above by the dimensionof the isometry group of the standard sphere and, on compact manifolds,equality is attained if and only if the manifold M is isometric to the standardsphere or the real projective space. Slightly more generally one canconsider conformal vector–fields, i.e., vector–fields with a flow preserving agiven conformal class of metrics. There are several geometrical conditionswhich force a conformal vector–field to be Killing [Semmelmann (2002)].A natural generalization of conformal vector–fields are the conformalKilling forms [Yano (1952)], also called twistor forms [Moroianu and Semmelmann(2003)]. These are p−forms α satisfying for any vector–field Xon the manifold M the Killing–Yano equation∇ X α − 1p+1 X ⌋ dα + 1n−p+1 X∗ ∧ d ∗ α = 0, (3.245)where n is the dimension of the manifold (M, g), ∇ denotes the covariantderivative of the Levi–Civita connection on M, X ∗ is 1−form dual to X and