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We prove a Bochner type vanishing theorem for compact complex manifolds $Y$in Fujiki class $\mathcal C$, with vanishing first Chern class, that admit acohomology class $[\alpha] \in H^{1,1}(Y,\mathbb R)$ which is numericallyeffective (nef) and has positive self-intersection (meaning $\int_Y \alpha^n\,>\, 0$, where $n\,=\,\dim_{\mathbb C} Y$). Using it, we prove that allholomorphic geometric structures of affine type on such a manifold $Y$ arelocally homogeneous on a non-empty Zariski open subset. Consequently, if thegeometric structure is rigid in the sense of Gromov, then the fundamental groupof $Y$ must be infinite. In the particular case where the geometric structureis a holomorphic Riemannian metric, we show that the manifold $Y$ admits afinite unramified cover by a complex torus with the property that the pulledback holomorphic Riemannian metric on the torus is translation invariant.

A Bochner principle and its applications to Fujiki class $\mathcal C$ manifolds with vanishing first Chern class