On the Index of a Fibered Manifold
S. S. Chern, F. Hirzebruch, J-P. Serre · Proceedings of the American Mathematical Society · 1957
Introduction.Let V be a real vector space of dimension r.Let F(x, y) = (x, y), x, y E V, be a real-valued symmetric bilinear function.We can find a base e,-, 1 ^i^r, in V, such that P P+3 (i) f(x, y) = zZ xy -zZ xy i-1 immp+1where x= zZt-i #'*< and y = X<-i y'**The number p-q is called the index of F, to be denoted by t(F).It depends only on F. If .F is nonsingular (i.e.p+q = r), then min (p, q) equals the maximal dimension of the linear subspaces of V contained