Let $M$ be a manifold. The total Stiefel-Whitney class of $M$ is defined to be the Stiefel-Whitney class of the tangent bundle $TM$
$$
w(M)=1+w\_1(TM)+w\_2(TM)+\cdots
$$
I want to find references for
$$
w(SO(n)/SO(k)),(i.e. w(V\_{n-k}(\mathbb{R}^n))), \\
w(U(n)/U(k)),(i.e. w(V\_{n-k}(\mathbb{C}^n))), \\
w(Sp(n)/Sp(k)),(i.e. w(V\_{n-k}(\mathbb{H}^n))),\\
w(SO(n)/(SO(k)\times SO(n-k))),(i.e. w(G\_{n-k}(\mathbb{R}^n))), \\
w(U(n)/(U(k)\times U(n-k))),(i.e. w(G\_{n-k}(\mathbb{C}^n))), \\
w(Sp(n)/(Sp(k)\times Sp(n-k))),(i.e. w(G\_{n-k}(\mathbb{H}^n))).
$$
Which ones of the above are known? Where could I find these formulas?



References for Stiefel-Whitney class of Stiefel manifolds and Grassmannians
The Stiefel manifolds are all parallelizable for $n-k\ge2$, so their total Stiefel-Whitney classes are equal to $1$. A reference is Theorem 3.1 of [this paper](http://www.ams.org/journals/tran/1975-213-00/S0002-9947-1975-0431194-X/home.html) by Kee Yuen Lam.


For the finite Grassmannians, things are a little more complicated. In the complex and symplectic cases you should be able to calculate these using the Wu formula. Partial results on the normal Stiefel-Whitney classes $\bar{w}(G\_{n-k}(\mathbb{R}^n))$ are scattered throughout the literature, a recent reference being 


Korbaš, J.; Novotny, P. [On the dual Stiefel-Whitney classes of some Grassmann manifolds](http://link.springer.com/article/10.1007%2Fs10474-008-8116-4). Acta Math. Hungar. 123 (2009), no. 4, 319–330.


This article and the articles it references are content with showing non-triviality of some normal SW-class in order to deduce non-immersion results, and do not give formulae for $w(G\_{n-k}(\mathbb{R}^n))$.


Most of the methods seem to use the vector bundle isomorphisms
\begin{align\*}
T(G\_{n-k}(\mathbb{R}^n)) & \cong \operatorname{Hom}(\gamma\_k,\gamma\_{n-k})\\
 & \cong \gamma\_k^\*\otimes \gamma\_{n-k} \\
 & \cong \gamma\_k\otimes \gamma\_{n-k}\end{align\*}
and the splitting principle.


