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.
Double forms, curvature structures and the $(p,q)$-curvatures
We introduce a natural extension of the metric tensor and the Hodge star operator to the algebra of double forms to study some aspects of the structure of this algebra. These properties are then used to study new Riemannian curvature invariants, called the (p, q)-curvatures. They are a generalization of the p-curvature obtained by substituting the Gauss-Kronecker tensor to the Riemann curvature tensor. In particular, for p = 0, the (0, q)-curvatures coincide with the H. Weyl curvature invariants, for p = 1 the (1, q)-curvatures are the curvatures of generalized Einstein tensors, and for q = 1 the (p, 1)-curvatures coincide with the p-curvatures. Also, we prove that the second H. Weyl curvature invariant is nonnegative for an Einstein manifold of dimension n > 4, and it is nonpositive for a conformally flat manifold with zero scalar curvature. A similar result is proved for the higher H. Weyl curvature invariants.
On the Bianchi identities
The object of this paper is to study the various curvatures which appear in differential geometry in different contexts. We shall point out the common analytic features underlying the tensors which appear in the definitions of these curvatures and equally emphasize the curvature functions themselves. Two basic examples are: 1. The second fundamental form of a hypersurface. The corresponding curvature function is the socalled normal curvature. 2. The Riemann curvature tensor of a Riemann manifold. The corresponding curvature function is sectional curvature. In [5-1 and [8] we have proved that under certain conditions a diffeomorphism of hypersurfaces preserving normal curvature is a congruence and a diffeomorphism of Riemann manifolds preserving sectional curvature is an isometry. Motivation of the present work was to seek generalization of these results to Gauss Kronecker curvatures. These curvatures arise as follows: the Gaussian definition of sectional curvature of a 2-dimensional surface via the Gauss map into the unit sphere can be imitated for a hypersurface in Euclidean space of arbitrary dimension. If the dimension of the hypersurface is even then the function obtained this way depends only on the intrinsic geometry of the hypersurface, and more generally for an arbitrary Riemann manifold (M, g) for each even positive integer 2q one can define a curvature function Kq which is a function on the Grassmann bundle of 2q-planes on the manifold. K 1 coincides with the sectional curvature. We shall call Kq = the q-th Gauss Kronecker curvature of M. It will be shown that /f dim M > 4q then generically a Kq-preserving diffeomorphism is an isometry (cf. Theorem 8.2). However this result is not as strong as in the case of sectional curvature. We have been able to carry through the algebraic part completely and in the process obtained some higher order presumably new conformal invariants. However the differential identities that we derive do not suffice to prove as strong a result as the one for sectional curvature. In [5-1 the result for sectional curvature preserving maps is proved completely only for dimension > 4. The case of 3-dimensional manifolds was largely unsettled. In this paper the proof of the result is greatly simplified and also two important cases when the dimension equals 3 are