A note on the characteristic rank of null-cobordant manifolds

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We give an upper bound for the characteristic rank of smooth closed null-cobordant manifolds and apply it to the Grassmann manifolds of oriented vector subspaces in Euclidean space. The bound turns out to be sharp: it coincides with the exact value of characteristic rank for the Grassmann manifolds of oriented 2-dimensional vector subspaces in odd-dimensional Euclidean space.