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where Qi5 denote the curvature forms and X(N) is the Euler-Poincare 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 (M2n-~1, g) of odd dimension, we have X (M) = 0. This shows that M = M21 admits a vector field with no singular points. If we try to find some formula on (M21, g) analogous to (1.1), some restriction on this may be necessary and it might be hoped that the right hand side is a linear combination of Betti numbers. We assume that e = eo is a unit vector field. Let wo be the 1-form dual to eo with respect to g. Then we have local fields of orthonormal vectors e0, e1, , e2n and the dual w0, w1, ... , wen. We call this frame field a e-frame field. By QAB (A, B =0, 1,..., 2n) we denote the curvature forms with respect to the above frame field. By 13r(M) we denote the r-th Betti number of M. In this paper we have THEOREM A. Let (M2n-~1, g) be a compact Riemannian manifold admitting a unit Killing vector e and let (e0, eti) be a e-frame field. Assume that

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