On the second Betti number of a compact Sasakian space

概要

論文の詳細を見る
Recently S.I. Goldberg [4] proved the following THEOREM A. If a compact, simply connected, regular 2m+1 dimensional Sasakian space has positive sectional curvature and its scalar curvature is constant, then it is isometric with a sphere S^<2m+1> with the natural structure. ?On the other hand the odd dimensional Betti number b_<2p+1>(M, R), 1≦2p+1≦m, of a compact Sasakian space M is even^1) and for the even dimensional Betti number of M the following theorem is known [4]. THEOREM B. If a compact, regular 2m +1 dimensional Sasakian space M has positive sectional curvature, then b_2(M, R)=0. ,The assumption "regular" in the theorems is essential, because the fibration of Boothby-Wang is used in their proofs. In this paper we shall prove the following theorem without the assumption "regular". THEOREM C. If any sectional curvature ρ(X, Y), of a complete 2m+1 (≧5). dimensional, Sasakian space M satisfies ρ(X, Y)>2m/1, then we have b_2(M, R)=0. REMARK. The metric of our Sasakian space is not normalized in the sence that the maximum sectional curvature is 1, though it has been normalized in a certain sence. As to the notations we follow S. Tachibana [5] and give definitions, preliminary facts and formulas in §1 and §2. In §3-§5 we shall prove Theorem C by the method of Berger [2] and Bishop-Goldberg [3].
お茶の水女子大学の論文