Existence of a certain kind of compact spin manifold with boundary

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.

• Welcome to mathoverflow! For question 1, if $\dim M$ is even, the spectrum is symmetric because $D$ anticommutes with the Clifford volume element. Have you tried to check wether the Clifford volume acts on the space of sections satisfying the APS conditions? – Sebastian Goette Dec 4 '18 at 15:35
• I thought about this and from the conversation I had with other people, it seems like the Clifford volume form $\omega^{\mathbb{C}}_n$ should acts on sections of spin structure on the boundary of $M$ (assuming dimension of $M$ is even) like $\text{identity} \oplus -\text{identity}$. If this is true, then the form should commute with the spectral projection, which in-turn preserves the APS boundary condition. – M. L. Nguyen Dec 5 '18 at 19:03