# Killing Spinors and Eigenvalues of Dirac Operators

Let $M$ be a compact n-manifold with chosen spin structure and corresponding Dirac operator $D$. We have Friedrich's sharp estimate $\lambda^2\ge \frac{R_0}{4}\frac{n}{n-1}$ for all eigenvalues of $D$, where $R_0$ is the minimum of the scalar curvature. Equality holds iff the corresponding eigenspinor $\psi_\lambda$ satisfies the field equation $\nabla_X\psi=\mp\sqrt\frac{R_0}{n(n-1)} X\cdot\psi$. In other words, $\psi_\lambda$ is a Killing spinor ($\exists\; c\in\mathbb{C}\;|\; \nabla_X\psi_\lambda=c X\cdot\psi_\lambda$).

When I apply this to some basic spaces, such spaces as $\mathbb{R}P^n$ which has two spin structures, I am lead to questions. For instance with $n=3$, a corresponding Dirac operator should have $\pm\frac{1}{2}\sqrt\frac{3}{2}$ as eigenvalues, but how can I guarantee the existence of Killing spinors? And does it work for both Dirac operators?
Is there another iff-statement for equality that helps us determine such lower-bound eigenvalues?

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