Scalar curvatures on $O(M),G_{2}(M)$

Abstract:

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