Ivancevic_Applied-Diff-Geom

. ..Applied Manifold Geometry 441intuition can be made precise, and one can indeed view the cohomologyclass of the resulting ‘delta–function’ p−form as the Poincar é dual to Σ.Going back to the relations (3.243), we see that the Hodge numbers of aKähler manifold can be nicely written in a so–called Hodge diamond form:h 0,0h 1,0 h 0,1. ..h m,0 · · · h 0,m. ..h m,m−1 h m−1,mh m,mThe integers in this diamond are symmetrical under the reflection in itshorizontal and vertical axes.. ..3.15 Conformal Killing–Riemannian GeometryIn this section we present some basic facts from conformal Killing–Riemannian geometry. In mechanics it is well–known that symmetries ofLagrangian or Hamiltonian result in conservation laws, that are used todeduce constants of motion for the trajectories (geodesics) on the configurationmanifold M. The same constants of motion are get using geometricallanguage, where a Killing vector–field is the standard tool for the descriptionof symmetry [Misner et al. (1973)]. A Killing vector–field ξ i is avector–field on a Riemannian manifold M with metrics g, which in coordinatesx j ∈ M satisfies the Killing equationξ i;j + ξ j;i = ξ (i;j) = 0, or L ξ ig ij = 0, (3.244)where semicolon denotes the covariant derivative on M, the indexed bracketdenotes the tensor symmetry, and L is the Lie derivative.The conformal Killing vector–fields are, by definition, infinitesimal conformalsymmetries i.e., the flow of such vector–fields preserves the conformalclass of the metric. The number of linearly–independent conformal Killingfields measures the degree of conformal symmetry of the manifold. Thisnumber is bounded by 1 2(n + 1)(n + 2), where n is the dimension of themanifold. It is the maximal one if the manifold is conformally flat [Baum(2000)].