The scalar curvature on totally geodesic fiberings
A compact Riemannian manifold \(N\) with scalar curvature \(\kappa _n\) is said to satisfy a comparison theorem for the scalar curvature iff for any other compact Riemannian manifold \(M\) (\(\dim M = \dim N\)) the inequality \(\kappa _M(x)\leq\kappa _N(f(x))\) holds at some \(x\in M\) whenever \(f:M\to N\) is a vector contracting spin map of non-zero degree. Rigidity holds if \(\kappa_M(x)\geq\kappa _N(f(x))\) for all \(x\) implies that \(f\) is an isometry. The main result reads as follows: If \(\pi :\tilde N\to N\) is a Riemannian submersion with totally geodesic fibres and \(\tilde N\) satisfies the above comparison theorem with rigidity, then \(N\) itself satisfies these conditions.
scientific article; zbMATH DE number 1563296