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What is the formal name of the following object: \begin{align}\tag{4} \Delta^{\alpha \beta} = \dfrac{\partial y^\alpha}{\partial x^m} g^{mn} \dfrac{\partial y^\beta}{\partial x^n} \end{align} where $g^{mn}$ is the inverse of the metric tensor $$ g_{mn} = \dfrac{\partial y^\alpha}{\partial x^m} \dfrac{\partial y_\alpha}{\partial x^n} $$ and $$ \dfrac{\partial y^\alpha}{\partial x^m} $$ is a Jacobian transformation from $x$ to $y$ coordinates. Willie Wong states that its related to the coordinate values of the pushforward of the inverse metric on an embedded manifold, however I am unsure if this is the formal name or not.

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    $\begingroup$ If $y(x)$ really is just a change of coordinates, then it's just the components of the inverse metric in the coordinate system $y$. $\endgroup$ Commented Jun 3, 2015 at 18:20
  • $\begingroup$ @AnthonyCarapetis Does it change its meaning if the dimension of $y$ is bigger than the dimension of $x$? $\endgroup$ Commented Jun 3, 2015 at 21:03

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If $y : M \to N$ is a map of smooth manifolds, then your coordinate expression for $\Delta$ defines a "tensor field along $y$" known as the pushforward of $g^{-1}$ by $y$, often notated by $y_* g^{-1}$. Thus "pushforward of the inverse metric" is an appropriate name. See e.g. this Wikipedia article for a description of the pushforward of vector fields - the situation here is identical in spirit.

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  • $\begingroup$ What is a pushforward? Is that a Jacobian transformation matrix? $\endgroup$ Commented Jun 4, 2015 at 1:15
  • $\begingroup$ @linuxfreebird: in general it's the idea of taking objects on the domain space and "pushing them on to" the target space using the map $y$ in the most natural way. For tensor fields this means applying the Jacobian to each index. $\endgroup$ Commented Jun 4, 2015 at 9:31

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