作者：Damianos Iosifidis
类似于非度量对偶连接的概念，这在定义统计流形中是必不可少的，我们发展了扭转对偶连接的定义。因此，我们说明了这种扭转对偶连接的几何意义，并展示了两种连接的使用如何在配有连接及其扭转对偶的空间中保持平行四边形的开裂。这种扭转对偶连接的系数基本上是通过要求两个连接之间的相互扭转消失来计算的。对于这个流形，我们证明了两个基本定理。特别是，如果两个连接都是度量兼容的，我们证明存在一个特定的$3$-形式，用于测量连接及其扭转对偶如何偏离Levi-Civita形式。此外，我们证明了对于这些扭转对偶流形，一个连接的平坦性不必将平坦性强加给另一个，而是后者的曲率张量由特定的散度给出。Finall公司
In analogy to the concept of a non-metric dual connection, which is essential in defining statistical manifolds, we develop that of a torsion dual connection. Consequently, we illustrate the geometrical meaning of such a torsion dual connection and show how the use of both connections preserves the cracking of parallelograms in spaces equipped with a connection and its torsion dual. The coefficients of such a torsion dual connection are essentially computed by demanding a vanishing mutual torsion among the two connections. For this manifold we then prove two basic Theorems. In particular, if both connections are metric-compatible we show that there exists a specific $3$-form measuring how the connection and its torsion dual deviate away from the Levi-Civita one. Furthermore, we prove that for these torsion dual manifolds flatness of one connection does not necessary impose flatness on the other but rather that the curvature tensor of the latter is given by a specific divergence. Finally, we give a self-consistent definition of the mutual curvature tensor of two connections and subsequently define the notion of a curvature dual connection.
论文链接：http://arxiv.org/pdf/2303.13259v1
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