Universal curvature identities

Abstract

We study scalar and symmetric 2-form valued universal curvature identities. We use this to establish the Gauss-Bonnet theorem using heat equation methods, to give a new proof of a result of Kuz'mina and Labbi concerning the Euler-Lagrange equations of the Gauss-Bonnet integral, and to give a new derivation of the Euh-Park-Sekigawa identity. MSC 2010: 53B20, 58G25.

A curvature identity on a~6-dimensional Riemannian manifold and its applications

“…We demonstrated that the obtained curvature identity holds on any 4-dimensional Riemannian manifold which is not necessarily compact [11]. Further, Gilkey, Park and Sekigawa extended the result to the higher dimensional setting, the pseudo-Riemannian setting, manifolds with boundary setting and the Kähler setting [13,14,15,16]. In this paper, we shall give a curvature identity explicitly which holds on any 6-dimensional Riemannian manifold using methods similar to those used in the 4-dimensional Chern-Gauss-Bonnet theorem and also provide some applications of the obtained curvature identity.…”

“…In this paper, we shall give a curvature identity explicitly which holds on any 6-dimensional Riemannian manifold using methods similar to those used in the 4-dimensional Chern-Gauss-Bonnet theorem and also provide some applications of the obtained curvature identity. More precisely, we derive a symmetric 2-tensor valued curvature identity of degree 6 which holds on any 5-dimensional Riemannian manifold, from which a scalar-valued curvature identity can be derived ( [13], Lemma 1.2 (3)). Furthermore, we derive a symmetric 2-tensor valued curvature identity of degree 6 on 4-dimensional Riemannian manifolds from the curvature identity on 5-dimensional Riemannian manifolds.…”

Curvature Identities Derived From an Integral Formula for the First Chern Number

“…Euh, Park and Sekigawa [3] gave a direct proof for Labbi's result in the 4-dimensional case and some applications of the curvature identity [4,5]. We refer [6] for the universality of the curvature identities in the Riemannian setting and we refer [7] for the universality of the curvature identities in the pseudo-Riemannain setting.…”

On some algebraic identities and the exterior product of double forms

“…The true question is then to determine the eigenvalues of N p (R) in terms of the [3,4] we define I p+1 m,n to be the space of invariant local formulas for symmetric (p, p) double forms that satisfy the first Bianchi identity and that are homogeneous of degree n in the derivatives of the metric and which are defined in the category of m dimensional Riemannian manifolds. In particular I …”

“…It the context of Riemannian geometry, where R is the Riemann curvature tensor (seen as a (2, 2) double form), the previous scalar curvature identities coincide with Gilkey-Park-Sekigawa universal curvature identities [4] which are shown to be unique. Also the identities of Proposition 3.5 coincide with the symmetric 2-form valued universal curvature identities of [4] where they are also be shown to be unique. The higher algebraic identities, that are under study here in this paper, can be seen then as symmetric double form valued universal curvature identities in the frame of Riemannian geometry.…”