On the span of the vector bundles m xi(sub n) over real projective spaces

Abstract

This paper is concerned with the maximal number (span) of linearly independent sections of the vector bundle m xi(sub n),which is defined as the m-fold Whitney sum of the canonical line bundle xi(sub n) over the n dimensional real projective space. The span of (n+1)xi(sub n) was proved to be 2(exp c) + 8d, when m = n + 1 was factorized as 2(sup c+4d) times the odd factor, where c is greater than or equal to zero and less than or equal to three, by the second author Yoshida. The purpose of this paper is to determine the necessary and sufficient conditions for span (n+r)xi(sub n) = k (r is greater than or equal to one, k is greater than or equal to one and less than or equal to eight). Firstly the known theorem are shown for the necessity of proof, and then the theorem of the necessary and sufficient conditions for span (n+r)xi(sub n) to be k is proved. Further the another proof of the theorem in the case of m = n + 1 is given.

本論文はn次元実射影空間内の標準的直線束ξ(sub n)のm重ホイットニー和で定義されるベクトル束mξ(sub n)の線形独立なセクションの最大数(スパン)を決定する問題を扱っている。著者の1人である 吉田は、m=n+1の場合について、n+1を因数分解したときの2のべき乗がc+4d(0≦c≦3、dは非負の整数)の時、(n+1)ξ(sub n)のスパンが2(exp c)+8dであることを証明した。本論文の目的は、span(n+r)ξ(sub n)=k(r≧1、1≦k≦8)になるための必要十分条件を決定することである。そのために必要なすでに知られている定理を示し、次に(n+r)ξ(sub n)のスパンについての必要十分条件に関する定理の証明を述べた。さらに(n+1)ξ(sub n)のスパンに関する定理について別の証明を与えた。