On some algebraic identities and the exterior product of double forms

Details On some algebraic identities and the exterior product of double forms On some algebraic identities and the exterior product of double forms

2011-12-06

arxiv.org/abs/1112.1346v2

Mathematics

Differential Geometry

26 pages, an error in lemma 1.2 corrected

Scientific paper

We use the exterior product of double forms to re-formulate celebrated classical results of linear algebra about matrices namely Cayley-Hamilton theorem, Laplace expansion of the determinant, Newton identities and Jacobi's formula for the determinant. This new formalism is then used to naturally generalize the previous results to double forms. In particular, we show that the Cayley-Hamilton theorem once applied to the second fundamental form of a hypersurface is equivalent to a linearized version of the Gauss-Bonnet 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 Gauss-Bonnet-Chern theorem. In addition to that, the general Cayley-Hamilton theorems generate several universal curvature identities in the sense of Gilkey-Park-Sekigawa. The extension of the classical Laplace expansion of the determinant to double forms is shown to lead to general Avez type formulas for all the Gauss-Bonnet curvatures.

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