REMARKS ON BIANCHI SUMS AND PONTRJAGIN CLASSES
REMARKS ON BIANCHI SUMS AND PONTRJAGIN CLASSES
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DOI：
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发表时间：
2014
影响因子：
0.7
通讯作者：
M. Labbi
中科院分区：
文献类型：
--
作者：
M. Labbi
Abstract We use the exterior and composition products of double forms together with the alternating operator to reformulate Pontrjagin classes and all Pontrjagin numbers in terms of the Riemannian curvature. We show that the alternating operator is obtained by a succession of applications of the first Bianchi sum and we prove some useful identities relating the previous four operations on double forms. As an application, we prove that for a $def xmlpi #1{}def mathsfbi #1{oldsymbol {mathsf {#1}}}let le =leqslant let leq =leqslant let ge =geqslant let geq =geqslant def Pr {mathit {Pr}}def Fr {mathit {Fr}}def Rey {mathit {Re}}k$-conformally flat manifold of dimension $ngeq 4k$, the Pontrjagin classes $P_i$ vanish for any $igeq k$. Finally, we study the equality case in an inequality of Thorpe between the Euler–Poincaré characteristic and the $k{
m th}$ Pontrjagin number of a $4k$-dimensional Thorpe manifold.
Abstract We use the exterior and composition products of double forms together with the alternating operator to reformulate Pontrjagin classes and all Pontrjagin numbers in terms of the Riemannian curvature. We show that the alternating operator is obtained by a succession of applications of the first Bianchi sum and we prove some useful identities relating the previous four operations on double forms. As an application, we prove that for a $def xmlpi #1{}def mathsfbi #1{oldsymbol {mathsf {#1}}}let le =leqslant let leq =leqslant let ge =geqslant let geq =geqslant def Pr {mathit {Pr}}def Fr {mathit {Fr}}def Rey {mathit {Re}}k$-conformally flat manifold of dimension $ngeq 4k$, the Pontrjagin classes $P_i$ vanish for any $igeq k$. Finally, we study the equality case in an inequality of Thorpe between the Euler–Poincaré characteristic and the $k{
m th}$ Pontrjagin number of a $4k$-dimensional Thorpe manifold.