Scalar curvature rigidity of hyperbolic product manifolds (Q1762709)

\textit{M. Min-Oo} [Math. Ann. 285, No. 4, 527--539 (1989; Zbl 0686.53038)] proved that a strongly asymptotically hyperbolic spin manifold \((M^n,g)\) with scalar curvature scal\(\geq -n(n-1)\) is isometric to the real hyperbolic space. The key points of this result are the existence of imaginary Killing spinors on the real hyperbolic space as well as the non-compact Bochner technique. In this paper a scalar rigidity result of the Riemannian product manifold \(\mathbb{R}^{m_1}\times \mathbb{R}H^{m_2}(-K_2)\times\dots \times\mathbb{R}H^{m_\ell}(-K_\ell)\) is given in analogy to M. Min-Oo's result, where \(\mathbb{R}H^{m_j}(-K_j)\) means the real hyperbolic space of dimension \(m_j\) and sectional curvature \(-K_j\). In order to prove this, the author considers Dirac bundles obtained from the spinor bundle, and derives Killing equations trivializing these Dirac bundles. Moreover, an integrated Bochner-Weitzenböck formula is shown which allows the usage of the non-compact Bochner technique.