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Related to the question Riemannian manifolds isometry, could anyone be able to explain to me what the notation $ \otimes $? Some examples were given in the question : $dx\otimes dx+dy\otimes dy = \frac12(dz\otimes d\bar z + d\bar z\otimes dz) = |dz|^2$ or $dw \otimes d\bar{w}$.

Clarification : I think my question is different of What does this symbol $\otimes$ mean?, because I would like some details on a specific example.

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David Labrecque
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1 Answers1

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It denotes a tensor product; I would do a no better job of describing it than the Wikipedia page: https://en.wikipedia.org/wiki/Tensor_product.

To understand tensor products from a linear algebra/module theory perspective I like Dummit and Foote's exposition. Formally the equality you have above comes from the parametrization $z= x+iy$, $\bar{z} = x -iy$, and billinearity of the tensor product.

GiantTortoise1729
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