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.
The Yamabe problem for higher order curvatures
Let M be a compact Riemannian manifold of dimension n > 2. The k-curvature, for k = 1,2,...,n, is defined as the k-th elementary symmetric polynomial of the eigenvalues of the Schouten tenser. The k-Yamabe problem is to prove the existence of a conformal metric whose k-curvature is a constant. When k = 1, it reduces to the well-known Yamabe problem. Under the assumption that the metric is admissible, the existence of solutions is known for the case k = 2, n = 4, for locally conformally flat manifolds and for the cases k > n/2. In this paper we prove the solvability of the k-Yamabe problem in the remaining cases k ≤ n/2, under the hypothesis that the problem is variational. This includes all of the cases k = 2 as well as the locally conformally flat case.
Sectional Curvatures and Characteristic Classes
We are concerned here with the relationship between the curvature properties of a riemannian manifold X and the global topological and differential invariants of X. An interesting result in this direction is Chern's theorem [6] that if X is compact, orientable, and has constant riemannian sectional curvature, then all Pontrjagin classes of X (with real coefficients) are zero. One of the main results in the present paper is that we can make weaker assumptions on the curvature of the manifold and still conclude that certain Pontrjagin classes must be zero. For each even integer p between 2 and the dimension of the manifold we define, after Allendoerfer [1], a smoothfunction yp on the Grassmann bundle of p-planes tangent to X. This function, called the pth sectional curvature of X, measures the Lipschitz-Killing curvature of geodesic p-dimensional submanifolds of X. The second sectional curvature 72 is the usual riemannian sectional curvature. The higher order sectional curvatures are weaker invariants of the riemannian structure than the riemannian sectional curvature, and it is in terms of these that we state our result mentioned above. The Pontrjagin classes we consider here are cohomology classes with real coefficients.