关于q-变形Levi-Civita联络
On q-deformed Levi-Civita connections
摘要：
我们探讨了在量子2-球和3-球上引入q-变形连接的可能性，满足与q-变形导子类似的扭曲莱布尼茨规则。我们证明了这样的联系总是存在于投射模上。此外，还引入了度量相容的条件，并给出了一个明确的公式，参数化了自由模上的所有度量联络。对于量子3-球面上的1-形式模，引入了q-变形的挠自由条件，并对满足一定实在条件的一般度量类导出了Levi-Civita联络的Christoffel符号的显式表达式.最后，我们构造了量子2-球面上一类投射模的度量联络。
英文摘要：
We explore the possibility of introducing q-deformed connections on the quantum 2-sphere and 3-sphere, satisfying a twisted Leibniz rule in analogy with q-deformed derivations. We show that such connections always exist on projective modules. Furthermore, a condition for metric compatibility is introduced, and an explicit formula is given, parametrizing all metric connections on a free module. For the module of 1-forms on the quantum 3-sphere, a q-deformed torsion freeness condition is introduced and we derive explicit expressions for the Christoffel symbols of a Levi-Civita connection for a general class of metrics satisfying a certain reality condition. Finally, we construct metric connections on a class of projective modules over the quantum 2-sphere.
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