Date: Wed, 26 Jan 2000 15:41:56 -0500 (EST)
From: "Tom Goodwillie,304 Kassar,863-2590,617-926-3565" <tomg@math.brown.edu>
Subject: Re: Orientability question

>  I would like to know what Stiefel and Grassmann manifolds are orientabel
> > and which are not.

I believe that the Grassmannian of k-planes in n-space is
non-orientable if and only if n is even and 0<k<n.

The Stiefel mflds are all orientable.

This all follows from the fact that if G is a connected Lie group and
H a closed subgroup then G/H is orientable if and only if the (adjoint)
action of H on the vector space g/h is orientation-preserving.

In the Stiefel cases G=SO(n) and H=SO(k). In the Grassmann cases
G=SO(n) and H is the intersection of G and O(k)xO(n-k).

 - Tom Goodwillie

P.S. This is the first time I have ever posted anything to a list
in my life. I hope I didn't mget the wrong answer.