Formal name for the coordinate values of the pushforward of the inverse metric on an embedded manifold?

Question

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.

Answer 1

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.