For a compact Riemannian rotary collector $ (M ^ n, g) $ *without* limit, $ n not equiv 3 mod 4 $, the Dirac operator associated with a fixed turn structure is well known. $ S rightarrow M $ have *real, discreet* spectrum and *symmetrical* around zero If $ (M ^ n, g) $ has a non-empty connected limit, for the Dirac operator spectrum $ D ^ S $ of a fixed turn structure $ S $ To be real and discreet, one has to subjugate the problem of self-worth to the Atiyah-Patodi-Singer (APS) condition. My first question is the following

**Question 1.** What kind of reasonable conditions (topological / geometric) do we have to impose in a compact turn? $ (M ^ n, g, partial M neq 0) $ or $ S $ so that the spectrum of the above mentioned. $ D ^ S $ Restricted to the APS condition is also symmetric with respect to zero, in addition to being real and discrete?

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

Yes $ E rightarrow (M ^ n, g, partial M neq 0) $ is any Hermitian package equipped with a compatible connection $ nabla ^ E $. It is not difficult to see that the twisted package $ S otimes E $ it's a Clifford package on $ M $, in which a globally defined notion of operator of associated generalized Dirac can be had $ D ^ {S} otimes E} $. At this point, we carefully consider a metric $ g $ in $ M $ so close $ partial M $ It seems $ M times[0r)$[0r)$[0r)$[0r)$. With such choice of $ g $, $ D ^ {S} otimes E} $ has an unmistakable induced Dirac operator in $ partial M $, denoted by $ D ^ {S} otimes E, partial} $. We say $ (M ^ n, g, partial M neq 0) $ is **CItd** If you have an Hermitian package $ (E, nabla ^ E) $ ($ E $ can depend on $ S $) such that $ D ^ {S} otimes E, partial} $ it is invertible

by $ n geq 4 $, we say $ M ^ n $ is a *good variety* If and only if $ M $ is **CSymm** and there is a smooth map $ f: M rightarrow mathbb {S} ^ n $ such that $ (M, f ^ {S} S $) $ make $ M $ a **CItd** manifold. here $ S_0 $ is *the* rotate the structure in $ mathbb {S} ^ n $. Here is my last question

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

If these questions turn out to be trivial, I would just like a clue or an intelligent observation that points me in the right direction so that I can continue on my own.