This is a crosspost from Stackexchange, where the question did not find any attention. This is probably due to the length of this post, but I did my best to organize it as well as I could. I am thankful for any answers.
I am currently reading Melrose's book "The Atiyah-Patodi-Singer Index Theorem", and I am somewhat stuck in the section where exact $b$-metrics are defined.
Let me briefly recall the relevant definitions: If $I_p$ denotes the ideal of smooth functions vanishing on $p \in X$, and $\mathcal{V}_b$ the set of vector fields on $X$ that are tangent to the boundary at the boundary, then the $b$-tangent space is defined by $$ ^bT_pX := \mathcal{V}_b \,/\, I_p \cdot \mathcal{V}_b$$ This is in fact a vector bundle over $X$ and a $b$-metric is just a smooth scalar product on this bundle.
Now an exact $b$-metric is a metric that can be written as $$ g = \bigl(\frac{dx}{x}\bigr)^2 + g^\prime,$$ for some boundary defining function $x$, where $g^\prime$ is a metric on the usual tangent space of $X$. (for this to make sense, one obviously has to observe that there is a natural map $^bT_pM \longrightarrow T_pM$ given by evaluation at $p$, and the above formula is understood in the sense that $g$ is the pullback of the right side along this map).
Now comes Exercise 2.9, and that is where my problem starts. The bundle $^bN\partial X$ is defined as the span of $x \frac{\partial}{\partial x}$ (which is a well-defined independent of $x$ on $\partial X$). This is a trivial line bundle over $\partial X$. Now (a bit implicitly), a closed $b$-metric is defined as a metric on $^bT X$ such that $$ g = \left(\frac{dx}{x} \right)^2 ~~ \text{when restricted to}~~^bN\partial X$$ and $$ \Gamma^\infty(\partial X, (^bN\partial X)^\perp) \subset \Gamma^\infty(\partial X, ^b TX) ~~~~ \text{is a Lie subalgebra},$$ where $(^b N\partial X)^\perp$ is the orthogonal complement of our line bundle above w.r.t. $g$.
Question 1: What is an example of a closed $b$-metric which is not exact?
It is claimed that $(^bN\partial X)^\perp \rightarrow X$ becomes a bunde with a flat connection, and that $g$ is exact if and only if this connection is a product connection.
Question 2: How can I construct this connection?
So far, I pretty much doubt that this assertion can be true in the first place. Let $X = B^n$, the $n$-ball. An exact $b$-metric is given by $$ g = \left(\frac{d x}{x}\right)^2 + g_{\mathrm{Std}},$$ where $x = 1-r$, $r$ being the distance function from the origin and of course $g_{\mathrm{Std}}$ is the standard metric. Now for all $p \in \partial X$, $$ ^bN\partial X_p = \{\alpha x \nu + x \mathcal{V}_b \mid \alpha \in \mathbb{R}\},$$ where $\nu$ is the (usual) normal field directed inward, and $$ (^bN\partial X_p)^\perp = \{V + x \mathcal{V}_b\},$$ where $V$ coincides near $p$ with a vector field on $S^{n-1}$ that is parallel translated inward. The footpoint evaluation gives a well-defined isomorphism of vector bundles from $(^bN\partial X)^\perp$ to $T \partial X$.
However, if for example $n=3$, then $\partial X = S^2$, so that $T\partial X \approx (^bN\partial X_p)^\perp$ does not admit a flat connection.
So probably
Question 3: What did I do wrong?
