For a compact spin Riemannian manifold $(M^n,g)$ *without* boundary, $n \not\equiv 3\mod 4$, it is well-known that the Dirac operator associated with a fixed spin structure $S\rightarrow M$ has *real, discrete* spectrum and *symmetric* about zero. In the case $(M^n,g)$ has non-empty connected boundary, for the spectrum of the Dirac operator $D^S$ of a fixed spin structure $S$ to be real and discrete, one has to subjugate the eigenvalue problem to the Atiyah-Patodi-Singer(APS) condition. My first question is as follows

Question 1.What sort of reasonable (topological/geometrical) conditions do we have to impose on compact spin $(M^n,g,\partial M \neq 0)$ or $S$ so that the spectrum of aforementioned $D^S$ restricted to APS condition is also symmetric about zero, besides being real and discrete?

If such a task for question 1 is possible, temporarily we shall call such manifold **CSymm**.

Let $E \rightarrow (M^n,g,\partial M\neq 0)$ be any Hermitian bundle equipped with a compatible connection $\nabla^E$. It is not hard to see that the twisted bundle $S \otimes E$ is a Clifford bundle over $M$, on which one can have a globally-defined notion of associated generalized Dirac operator $D^{S\otimes E}$. At this point, we carefully consider a metric $g$ on $M$ so that near $\partial M$ looks like $\partial M \times [0,r)$. With such a choice of $g$, $D^{S\otimes E}$ has a unambiguous induced Dirac operator on $\partial M$, denoted by $D^{S\otimes E, \partial}$. We say $(M^n,g,\partial M \neq 0)$ is **CitD** iff it possesses a Hermitian bundle $(E,\nabla^E)$ ($E$ may depend on $S$) such that $D^{S\otimes E,\partial}$ is invertible.

For $n\geq 4$, we say $M^n$ is a *good manifold* if and only if $M$ is **CSymm** and there exists a smooth map $f: M \rightarrow \mathbb{S}^n$ such that $(M,f^{*} S_0)$ makes $M$ a **CitD** manifold, where $S_0$ is *the* spin structure on $\mathbb{S}^n$. Here is my last question

Question 2. What is an example of a good manifold with positive scalar curvature?

If these questions turn out to be straight-up trivial, I would just like a hint or a smart observation that would point me in a right direction so that I can go on on my own.