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Harmonic and killing vectors in a compact Riemannian manifold were studied by Yano & Bochner in [1]. In the presen paper a definition is given for harmonic and killing vectors of type p which, when p = 1, reduce to harmonic and killing vectors as given in [1]. Properties of harmonic and killing vectors of type p are obtained which correspond to harmonic and killing vectors (of type one). In the last two sections relations between harmonic and killing vectors of type p + 1 and of type p have been established in an orientable maniforld.

Asymptotic lines of order p have been defined by Hayden in [1]. In this paper we define geodesies of order p, investigate their properties and establish their relationship with the asymptotic lines of order p.