A formula on some odd-dimensional Riemannian manifolds related to the Gauss-Bonnet formula

where $\Omega_{ij}^{\prime}$ denote the curvature forms and $\chi(N)$ is the Euler-Poincar\’e characteristic. The left hand side of (1.1) is a differential geometric or Riemannian geometric quantity and the right hand side is a topological quantity. In (1.1), even dimensionality is essential. For a compact orientable Riemannian manifold $(M^{2n+1}, g)$ of odd dimension, we have $\chi(M)=0$ . This shows that $M=M^{2n+1}$ admits a vector field $\xi$ with no singular points. If we try to find some formula on $(M^{2n+1}, g)$ analogous to (1.1), some restriction on this $\xi$ may be necessary and it might be hoped that the right hand side is a linear combination of Betti numbers. We assume that $\xi=e_{0}$ is a unit vector field. Let $w_{0}$ be the l-form dual to $e_{0}$ with respect to $g$. Then we have local fields of orthonormal vectors