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On some algebraic identities and the exterior product of double forms
 ARCHIVUM MATHEMATICUM
, 2013
"... We use the exterior product of double forms to free from coordinates celebrated classical results of linear algebra about matrices and bilinear forms namely CayleyHamilton theorem, Laplace expansion of the determinant, Newton identities and Jacobi’s formula for the determinant. This coordinate fre ..."
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We use the exterior product of double forms to free from coordinates celebrated classical results of linear algebra about matrices and bilinear forms namely CayleyHamilton theorem, Laplace expansion of the determinant, Newton identities and Jacobi’s formula for the determinant. This coordinate free formalism is then used to easily generalize the previous results to higher multilinear forms namely to double forms. In particular, we show that the CayleyHamilton theorem once applied to the second fundamental form of a hypersurface is equivalent to a linearized version of the GaussBonnet theorem, and once its generalization is applied to the Riemann curvature tensor (seen as a (2, 2) double form) is an infinitisimal version of the general GaussBonnetChern theorem. In addition to that, we show that the general CayleyHamilton theorems generate several universal curvature identities. The generalization of the classical Laplace expansion of the determinant to double forms is shown to lead to new general Avez type formulas for all GaussBonnet curvatures.
On Gauss–Bonnet Curvatures
 SYMMETRY, INTEGRABILITY AND GEOMETRY: METHODS AND APPLICATIONS
, 2007
"... The (2k)th Gauss–Bonnet curvature is a generalization to higher dimensions of the (2k)dimensional Gauss–Bonnet integrand, it coincides with the usual scalar curvature for k = 1. The Gauss–Bonnet curvatures are used in theoretical physics to describe gravity in higher dimensional space times where ..."
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The (2k)th Gauss–Bonnet curvature is a generalization to higher dimensions of the (2k)dimensional Gauss–Bonnet integrand, it coincides with the usual scalar curvature for k = 1. The Gauss–Bonnet curvatures are used in theoretical physics to describe gravity in higher dimensional space times where they are known as the Lagrangian of Lovelock gravity, Gauss–Bonnet Gravity and Lanczos gravity. In this paper we present various aspects of these curvature invariants and review their variational properties. In particular, we discuss natural generalizations of the Yamabe problem, Einstein metrics and minimal submanifolds. Key words: Gauss–Bonnet curvatures; Gauss–Bonnet gravity; lovelock gravity; generalized Einstein metrics; generalized minimal submanifolds; generalized Yamabe problem 2000 Mathematics Subject Classification: 53C20; 53C25 1 An introduction to Gauss–Bonnet curvatures We shall present in this section several approaches to the Gauss–Bonnet curvatures. For precise definitions and examples the reader is encouraged to consult [17, 18, 20]. 1.1 Gauss–Bonnet curvatures vs. curvature invariants of Weyl’s tube formula In a celebrated paper [38] published in 1939, Hermann Weyl proved that the volume of a tube of radius r around an embedded compact psubmanifold M of the ndimensional Euclidean space is a polynomial in the radius of the tube as follows: Vol(tube(r)) = [p/2] i=0