Scalar curvatures on 𝑂(𝑀),𝐺₂(𝑀)

Abstract

We show that every C ∞ f : G 2 ( M ) → R , M n {C^\infty }f:{G_2}(M) \to {\mathbf {R}},{M^n} a compact connected riemannian manifold n ⩾ 3 n \geqslant 3 , is the scalar curvature function of some complete riemannian metric on G 2 ( M ) {G_2}(M) , the grassmann bundle of 2 2 planes over M M , except possibly when K = constant  ⩾ 0 K = {\text {constant }} \geqslant 0 . A similar result holds for O ( M ) O(M) bundle of orthonormal frames on M M .