In this paper [1] the author wrote and interresting relationship between differential geometry objects (the Levi-civita connection and Koszul form) by means of a musical isomorphism [2] (roughly speaking, the operation of Rasing and Lowering indexes of tensors [3]):

$$\nabla_{X}Y = \mathcal{K}(X,Y,\cdot)^{\#} \tag{1}$$

Well, this relationship is somekind of theorem? (I did a quick search and I didn't found anything)

$$* * *$$

[1] https://arxiv.org/abs/1207.5303

[2] https://en.wikipedia.org/wiki/Musical_isomorphism

[3] https://en.wikipedia.org/wiki/Raising_and_lowering_indices


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