Ask for two symbols

Question

I find several symbols in Differentiable Manifold. I list them in the following, $$ \int_{\Omega}*(V)^b. $$ I wanna ask about the meaning of $V^b$ so that I can calculate it.

For $*$, I guess it means the dual space of $V^b$.

Answer 1

If $V$ is a vector space, $V^*$ is often used to denote the dual space, but in this case, the thing in the integral must be a differential form, so "$*$" probably means something like the Hodge-star operator. The superscript "b" doesn't mean anything to me except that maybe it means "apply hodge star b times," but that doesn't really make sense. More context is needed.

Answer 2

Well, I found an answer in question 2683129, which says that:

You can avoid this problem using the musical isomorphism $\flat$, which is a map between the tangent bundle of $\mathbb{R}^3$ and its cotangent. If this does not sound familiar, you may think that $\flat$ takes a vector field on $\mathbb{R}^3$ in input, and returns a dual form defined on $\mathbb{R}^3$. $\flat$ is defined through the metric $g$ as follows: $\flat(X) \mapsto X^{\flat}$, where $X^{\flat}$ is a 1-form now, and it acts on vector fields like this: $X^{\flat}(Y) = g(X,Y)$. Using this definition you can compute $V^{\flat} = xdz + ydx$. In fact $V^{\flat}(V) = x^2+y^2 = g(V,V)$. Further, if $U = \sum_{i=1}^3 f_i(x,y,z)\frac{\partial}{\partial x^i}$, then $V^{\flat}(U) = g(V,U)$ (check!).

This paragraph may help me.