On a variational formula for the H. Weyl curvature invariants
We use the ring of curvature structures, to answer a question of M. Berger about a variational formula for the H. Weyl curvature invariants. Recall that the (2k)-H. Weyl curvature invariant is a polynomial of degree 2k with respect to the Riemannian curvature. In dimension 2k it is just the Gauss-Bonnet integrand, in higher dimensions it appears naturally, as an integrand, in the well known H. Weyl’s tube formula. Mathematics Subject Classification (2000). 58E11, 58C99.
Renormalizing curvature integrals on Poincaré–Einstein manifolds
Abstract After analyzing renormalization schemes on a Poincare–Einstein manifold, we study the renormalized integrals of scalar Riemannian invariants. The behavior of the renormalized volume is well known and we show any scalar Riemannian invariant renormalizes similarly. We consider characteristic forms and their behavior under a variation of the Poincare–Einstein structure and obtain, from the renormalized integral of the Pfaffian, an extension of the Gauss–Bonnet theorem.
A CLASS OF ALMOST CONTACT RIEMANNIAN MANIFOLDS
Recently S. Tanno has classified connected almostcontact Riemannian manifolds whose automorphism groups have themaximum dimension [9]. In his classification table the almost contactRiemannian manifolds are divided into three classes: (1) homogeneousnormal contact Riemannian manifolds with constant 0-holomorphic sec-tional curvature if the sectional curvature for 2-planes which contain