论文标题
Levi-Civita连接和非交通差分计算的矢量场
Levi-Civita connections and vector fields for noncommutative differential calculi
论文作者
论文摘要
We study covariant derivatives on a class of centered bimodules $\mathcal{E}$ over an algebra A. We begin by identifying a $\mathbb{Z} ( A ) $-submodule $ \mathcal{X} ( A ) $ which can be viewed as the analogue of vector fields in this context; $ \ Mathcal {x}（a）$被证明是谎言代数。 $ \ Mathcal {E} $上的连接是与$ \ Mathcal {x}（a）上的协变量的一对一信件。 $我们恢复了以协变量形式的连接的扭转和度量兼容性的经典公式。结果，还得出了Levi-Civita连接的Koszul公式。
We study covariant derivatives on a class of centered bimodules $\mathcal{E}$ over an algebra A. We begin by identifying a $\mathbb{Z} ( A ) $-submodule $ \mathcal{X} ( A ) $ which can be viewed as the analogue of vector fields in this context; $ \mathcal{X} ( A ) $ is proven to be a Lie algebra. Connections on $\mathcal{E}$ are in one to one correspondence with covariant derivatives on $ \mathcal{X} ( A ). $ We recover the classical formulas of torsion and metric compatibility of a connection in the covariant derivative form. As a result, a Koszul formula for the Levi-Civita connection is also derived.
