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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.
COMPACT MANIFOLDS WITH POSITIVE Γ2CURVATURE
"... Abstract. The Schouten tensor A of a Riemannian manifold (M, g) provides important scalar curvature invariants σk, that are the symmetric functions on the eigenvalues of A, where, in particular, σ1 coincides with the standard scalar curvature Scal(g). Our goal here is to study compact manifolds with ..."
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Abstract. The Schouten tensor A of a Riemannian manifold (M, g) provides important scalar curvature invariants σk, that are the symmetric functions on the eigenvalues of A, where, in particular, σ1 coincides with the standard scalar curvature Scal(g). Our goal here is to study compact manifolds with positive Γ2curvature, i.e., when σ1(g)> 0 and σ2(g)> 0. In particular, we prove that a 3connected nonstring manifold M admits a positive Γ2curvature metric if and only if it admits a positive scalar curvature metric. Also we show that any finitely presented group pi can always be realised as the fundamental group of a closed manifold of positive Γ2curvature and of arbitrary dimension greater than or equal to six.