Some Remarks on Tensor Fields and Curvature

S. Bochner [1, 2, 3] has recently developed a beautiful theory on curvature and Betti numbers of an orientable compact Riemannian space V. with positive definite metric. He starts from the lemma: In V, , if the Laplacian Azp= gt'j;j; j _ 0 everywhere for a certain scalar (p, then we have Asp = 0 everywhere. In the above statement, the semi-colon denotes the covariant differentiation. He then takes a harmonic tensor tiji2...ip and put (p = ilit2 ...ilt2. ..ip . He calculates the Azp and seeks for the conditions on curvature that make the Laplacian Azp ? 0. Thus, under certain conditions, he concludes the non-existence of harmonic tensor and consequently the vanishing of the Betti numbers by virtue of a theorem of Hodge [6]. In a recent paper [11], the present author has proved an integral formula