Fedosov和Riemann超流形
Fedosov and Riemannian supermanifolds
摘要：
分析了超流形的辛结构和度量结构的推广。根据相应的二次张量的偶/奇性质，两种类型的结构是可能的。在偶数的情况下，有一组非常丰富的几何结构：偶数辛超流形（或等价地，具有非退化泊松结构的超流形），偶数费多索夫超流形和甚至黎曼超流形。详细分析了这些结构之间存在的联系。在奇的情况下，我们证明了奇黎曼和费多索夫超流形的特征在于一个标量曲率张量。然而，奇黎曼超流形只能有常曲率。
英文摘要：
Generalizations of symplectic and metric structures for supermanifolds are analyzed. Two types of structures are possible according to the even/odd character of the corresponding quadratic tensors. In the even case one has a very rich set of geometric structures: even symplectic supermanifolds (or, equivalently, supermanifolds with non-degenerate Poisson structures), even Fedosov supermanifolds and even Riemannian supermanifolds. The existence of relations among those structures is analyzed in some details. In the odd case, we show that odd Riemannian and Fedosov supermanifolds are characterized by a scalar curvature tensor. However, odd Riemannian supermanifolds can only have constant curvature.
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