Showing papers in "Hokkaido Mathematical Journal in 1978"

On infinitesimal projective transformations satisfying the certain conditions

Abstract: We consider the following problem PROBLEM. Let M be a compact Riemannian manifold with positive constant scalar curvature. If M adminits a nonisometric infifinitesimal prOjective transformation, then is M a space of positive constant curvature? For this problem, the following results are known. THEOREM A. Let M be a complete Riemannian manifold with parallel Ricci tensor. IfM admits nonajfine infifinitesimal projective transformations, then M is a space of positive constant curvature. [1]. THEOREM B. Let M be a compact Riemannian manifold with consant scalar curvature K. If the scalar curvature is nonpositive, then an infifinitesimal projective transformation is a motion. [2]. THEOREM C. Let M be a compact Riemannian manifold satisfying a condition \ abla_{k}K_{fi}-\ abla_{f}K_{ki}=0, (K\ eq 0), where \ abla_{k} , K_{fi} denote a covariant derivative and Ricci tensor, respectively. The projective Killing vector v^{h} can be decomposed uniquely as follows,

On tensor products of linear operators

Abstract: \\otimes D[B] . The aim of this note is to study the closability of A\\otimes B in X\\otimes_{\\alpha}^{\\wedge}Y, and the range and null space of its closure A\\wedge\\otimes_{\\alpha}B if closable. The results will amplify [11, Theorem 4. 2] and [12, Theorems 4. 4, 4. 5 and 4. 6], which have been concerned with the problem of when A\\otimes_{\\alpha}B\\wedge-\\lambda is Fredholm or semi-Fredholm except for \\lambda=0 . We shall use the notions of reasonable norms, \\otimes -norms, (leftand right-) injective and projective \\otimes -norms in the sense of Grothendieck [6]. The greatest reasonable norm \\pi is a projective \\otimes -norm and the smallest reasonable norm \\epsilon an injective \\otimes -norm. Schatten [14] denoted them by