Graded metrics adapted to splittings
Graded metrics adapted to splittings
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适应分裂的分级指标
DOI：
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发表时间：
1997
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影响因子：
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通讯作者：
O. Sánchez
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文献类型：
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作者：
J. Monterde;O. Sánchez
本文完整地描述了在Koszul最近提出的意义下，其Levi-Civita连接适应于给定分裂的可裂分次流形上的齐次分次度量.其中的一个子类是通过渐变曲率张量的某些分量的消失而被挑选出来的，这个条件起着类似于渐变辛几何中渐变辛形式的封闭性的作用：它相当于由数据{g，ω，Δ′}确定一个分次度量，其中g是M上的度量张量，ω 0是Batchelor矩阵E → M上的纤维非退化反对称双线性型，并且Δ′是满足Δ′ω = 0的E上的联络。奇度量也在相同的准则下进行了研究，它们由数据{κ，Δ′}来指定，其中κ ∈ Hom（TM，E）可逆，Δ′κ = 0。一般地，证明了具有常分次曲率的偶分次度量只能在常曲率黎曼流形上支持，且Δ′ onE的曲率满足R Δ′（X，Y）2 = 0.证明了分次Ricci平坦偶度量在Ricci平坦流形上是可支撑的，并且联络Δ′的曲率满足一个特定的方程组. 0最后，分次Einstein偶度量只能在Ricci平坦黎曼流形上得到支持。讨论了Ω（M）上分次度量的相关结果.
Homogeneous graded metrics over split ℤ2-graded manifolds whose Levi-Civita connection is adapted to a given splitting, in the sense recently introduced by Koszul, are completely described. A subclass of such is singled out by the vanishing of certain components of the graded curvature tensor, a condition that plays a role similar to the closedness of a graded symplectic form in graded symplectic geometry: It amounts to determining a graded metric by the data {g, ω, Δ′}, whereg is a metric tensor onM, ω 0 is a fibered nondegenerate skewsymmetric bilinear form on the Batchelor bundleE → M, and Δ′ is a connection onE satisfying Δ′ω = 0. Odd metrics are also studied under the same criterion and they are specified by the data {κ, Δ′}, with κ ∈ Hom (TM, E) invertible, and Δ′κ = 0. It is shown in general that even graded metrics of constant graded curvature can be supported only over a Riemannian manifold of constant curvature, and the curvature of Δ′ onE satisfiesRΔ′ (X,Y)2 = 0. It is shown that graded Ricci flat even metrics are supported over Ricci flat manifolds and the curvature of the connection Δ′ satisfies a specific set of equations. 0 Finally, graded Einstein even metrics can be supported only over Ricci flat Riemannian manifolds. Related results for graded metrics on Ω(M) are also discussed.