Chapter VIII Conformal vector fields on the unitary tangent fibre bundle

Abstract

Let X be a vector field over M and exp(tX) the local I-parameter group generated by X and exp t X ^ its lift to V(M). We call X a conformal vector field or a conformal infinitesimal transformation if there exists a function φ on M such that the Lie derivative of the metric tensor g is equal to 2 φ.g. To every field of co-vectors Y(z) ∈ A 1 (W) is associated a covariant symmetric 2-tensor τ (Y). We calculate its square by making intervene explicitly the curvature of the space. This formula helps us characterize the infinitesimal conformal transformations when M is compact without boundary. In paragraph 4 we establish a formula giving the square of the vertical part of the lift of -conformal vector field on V(M) in terms of the flag curvature and we show that there exists a non-trivial conformal vector field only if the integral of the quadratic form (R(X, u)u, X) is positive. Next we take up the case when the scalar curvature H ˜ = g jk H ˜ jk where 2 H ˜ jk = ∂ 2 H v , v / ∂ v j ∂ k est a non-positive constant ( H ˜ = constant ≤ 0.) and the torsion trace co-vector satisfies a certain condition. On making a hypothesis on the vertical Ricci curvature Q ij we show that for dim M > 2, and M compact, the largest connected group C 0 (M) of infinitesimal conformal transformations coincides with the largest connected group of isometry I 0 (M), thus generalizing the Riemannian case. We deal with case where the l-form X = X i (z)dx i corresponding to the vector field is semi-closed and obtain in the conformal case the corresponding equations. Let σ be a semi-closed l-form whose co-differential δσ is independent of the direction and let Δ ¯ σ be the horizontal Laplacian and let λ (y), y ∈ W(M) a the function such that Δ ¯ σ = λ y σ . We give an estimate of λ(y) as a function of proper value of the flag curvature. The rest of the chapter is devoted to the lift to W(M) of the conformal vector field when its dual l-form is semi-closed.