光滑无穷维Grassmann流形G(E)(英文)

INFINITE DIMENSIONAL GRASSMANN MANIFOLDS G(E)

设E是Banach空间,G(E)表示空间E中可分裂的子空间全体,和U(N)={HE:H  N=E}.让F∈U(N).1983年,Abraham,Marsden和Ratiu给出G(E)上的一个微分结构{(U(N),Ψ_(F,N)}_(N∈G(E)),使得G(E)成为光滑Banach流形.然而即使在1988年他们的第二版书中,粘贴映射Ψ_(F,N) oΨ_(F_1,N_1)^(-1)光滑性的证明有洞,仍未成功.在这篇小文里,这个洞被指出,并给出了一个新的证明.这样,光滑无穷维Grassmann流形G(E)完全被证明.

Let E be a Banach space,G(E) the set of all split subspaces in E,and U(N)={H E:H N = E}.In 1983,Abraham,Marsden and Ratiu first proposed a differentiable structure on G(E),{(U(N),ψ_(F,N))}_(N∈G(E)) where F  N = E.It makes G(E) become a infinite dimensional and smooth Banach manifold.As we know,for the projective spaces RP^n and CP^n,the smooth infinite dimensional manifold G(E) will play an important role in the study of infinite dimensional geometry.However,even in 1988,the proof of the smooth overlap m...