Geodesic
On some algebraic identities and the exterior product of double forms
Archivum mathematicum, Tome 49 (2013) no. 4, pp. 241-271 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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 Cayley-Hamilton 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 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, we show that the general Cayley-Hamilton 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 Gauss-Bonnet curvatures.
We use the exterior product of double forms to free from coordinates celebrated classical results of linear algebra about matrices and bilinear forms namely Cayley-Hamilton 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 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, we show that the general Cayley-Hamilton 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 Gauss-Bonnet curvatures.
DOI : 10.5817/AM2013-4-241
Classification : 15A24, 15A63, 15A75, 15Axx, 53B20
Keywords: Cayley-Hamilton theorem; cofactor; characteristic coefficients; Laplace expansion; Newton identities; Jacobi’s formula; double form; Newton transformation; exterior product; Gauss-Bonnet theorem 
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Labbi, Mohammed Larbi. On some algebraic identities and the exterior product of double forms. Archivum mathematicum, Tome 49 (2013) no. 4, pp. 241-271. doi: 10.5817/AM2013-4-241

[1] Barvinok, A. I.: New algorithms for linear $k$-matroid intersection and matroid $k$-parity problems. Math. Programming 69 (1995), 449–470. | DOI | Zbl

[2] Gårding, L.: Linear hyperbolic differential equations with constant coefficients. Acta Math. 85 (1951), 2–62. | DOI

[3] Gilkey, P.: Invariance theory, the heat equation and the Atiyah-Singer index theorem. 2nd ed., CRC Press, 1994.

[5] Greub, W. H.: Multilinear algebra. 2nd ed., Springer-Verlag, New York, 1978. | Zbl

[7] Kulkarni, R. S.: On the Bianchi Identities. Math. Ann. 199 (1972), 175–204. | DOI | Zbl

[8] Labb, M. L.: Variational properties of the Gauss-Bonnet curvatures. Calc. Var. Partial Differential Equations 32 (2008), 175–189. | DOI | MR

[9] Labbi, M. L.: Double forms, curvature structures and the $(p,q)$-curvatures. Trans. Amer. Math. Soc. 357 (10) (2005), 3971–3992. | DOI | MR | Zbl

[10] Labbi, M. L.: On Weitzenböck curvature operators. arXiv:math/0607521v2 [math.DG], 2006.

[11] Labbi, M. L.: About the $h_{2k}$ Yamabe problem. arXiv:0807.2058v1 [math.DG], 2008.

[12] Labbi, M. L.: On $2k$-minimal submanifolds. Results Math. 52 (3–4) (2008), 323–338. | DOI | MR | Zbl

[13] Labbi, M. L.: Remarks on generalized Einstein manifolds. Balkan J. Geom. Appl. 15 (2) (2010), 61–69. | MR

[14] Luque, J. G., Thibon, J. Y.: Pfaffian and Hafnian identities in shuffle algebras. Adv. in Appl. Math. 29 (4) (2002), 620–646. | DOI | MR | Zbl

[15] Reilly, R. C.: Variational properties of functions of the mean curvatures for hypersurfaces in space forms. J. Differential Geom. 8 (1973), 465–477. | Zbl

[16] Vanstone, J. R.: The Poincaré map in mixed exterior algebra. Canad. Math. Bull. 26 (2) (1983). | DOI | Zbl

[17] Winitzki, S.: Linear Algebra via Exterior Products. GNU Free Documentation License, 2010.

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